Precalculus Examples

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$ , $y + 49 + \frac{5 x - 50}{y} = 0$

Step 1

Solve for $x$ in $y + 49 + \frac{5 x - 50}{y} = 0$ .

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Step 1.1

Simplify $y + 49 + \frac{5 x - 50}{y}$ .

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Step 1.1.1

Factor $5$ out of $5 x - 50$ .

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Step 1.1.1.1

Factor $5$ out of $5 x$ .

$y + 49 + \frac{5 (x) - 50}{y} = 0$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 1.1.1.2

Factor $5$ out of $- 50$ .

$y + 49 + \frac{5 x + 5 \cdot - 10}{y} = 0$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 1.1.1.3

Factor $5$ out of $5 x + 5 \cdot - 10$ .

$y + 49 + \frac{5 (x - 10)}{y} = 0$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

$y + 49 + \frac{5 (x - 10)}{y} = 0$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 1.1.2

To write $y$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$y \cdot \frac{y}{y} + \frac{5 (x - 10)}{y} + 49 = 0$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 1.1.3

Simplify terms.

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Step 1.1.3.1

Combine $y$ and $\frac{y}{y}$ .

$\frac{y \cdot y}{y} + \frac{5 (x - 10)}{y} + 49 = 0$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 1.1.3.2

Combine the numerators over the common denominator.

$\frac{y \cdot y + 5 (x - 10)}{y} + 49 = 0$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

$\frac{y \cdot y + 5 (x - 10)}{y} + 49 = 0$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 1.1.4

Simplify the numerator.

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Step 1.1.4.1

Multiply $y$ by $y$ .

$\frac{y^{2} + 5 (x - 10)}{y} + 49 = 0$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 1.1.4.2

Apply the distributive property.

$\frac{y^{2} + 5 x + 5 \cdot - 10}{y} + 49 = 0$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 1.1.4.3

Multiply $5$ by $- 10$ .

$\frac{y^{2} + 5 x - 50}{y} + 49 = 0$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

$\frac{y^{2} + 5 x - 50}{y} + 49 = 0$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 1.1.5

To write $49$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{y^{2} + 5 x - 50}{y} + 49 \cdot \frac{y}{y} = 0$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 1.1.6

Combine $49$ and $\frac{y}{y}$ .

$\frac{y^{2} + 5 x - 50}{y} + \frac{49 y}{y} = 0$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 1.1.7

Combine the numerators over the common denominator.

$\frac{y^{2} + 5 x - 50 + 49 y}{y} = 0$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

$\frac{y^{2} + 5 x - 50 + 49 y}{y} = 0$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 1.2

Set the numerator equal to zero.

$y^{2} + 5 x - 50 + 49 y = 0$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 1.3

Solve the equation for $x$ .

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Step 1.3.1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.3.1.1

Subtract $y^{2}$ from both sides of the equation.

$5 x - 50 + 49 y = - y^{2}$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 1.3.1.2

Add $50$ to both sides of the equation.

$5 x + 49 y = - y^{2} + 50$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 1.3.1.3

Subtract $49 y$ from both sides of the equation.

$5 x = - y^{2} + 50 - 49 y$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

$5 x = - y^{2} + 50 - 49 y$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 1.3.2

Divide each term in $5 x = - y^{2} + 50 - 49 y$ by $5$ and simplify.

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Step 1.3.2.1

Divide each term in $5 x = - y^{2} + 50 - 49 y$ by $5$ .

$\frac{5 x}{5} = \frac{- y^{2}}{5} + \frac{50}{5} + \frac{- 49 y}{5}$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 1.3.2.2

Simplify the left side.

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Step 1.3.2.2.1

Cancel the common factor of $5$ .

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Step 1.3.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- y^{2}}{5} + \frac{50}{5} + \frac{- 49 y}{5}$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 1.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- y^{2}}{5} + \frac{50}{5} + \frac{- 49 y}{5}$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

$x = \frac{- y^{2}}{5} + \frac{50}{5} + \frac{- 49 y}{5}$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

$x = \frac{- y^{2}}{5} + \frac{50}{5} + \frac{- 49 y}{5}$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 1.3.2.3

Simplify the right side.

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Step 1.3.2.3.1

Simplify each term.

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Step 1.3.2.3.1.1

Move the negative in front of the fraction.

$x = - \frac{y^{2}}{5} + \frac{50}{5} + \frac{- 49 y}{5}$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 1.3.2.3.1.2

Divide $50$ by $5$ .

$x = - \frac{y^{2}}{5} + 10 + \frac{- 49 y}{5}$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 1.3.2.3.1.3

Move the negative in front of the fraction.

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y + x^{2} - 5 x - 50 = 0$

Step 2

Replace all occurrences of $x$ with $- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $y^{2} + 49 y + x^{2} - 5 x - 50 = 0$ with $- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$ .

$y^{2} + 49 y + {(- \frac{y^{2}}{5} + 10 - \frac{49 y}{5})}^{2} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $y^{2} + 49 y + {(- \frac{y^{2}}{5} + 10 - \frac{49 y}{5})}^{2} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(- \frac{y^{2}}{5} + 10 - \frac{49 y}{5})}^{2}$ as $(- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5})$ .

$y^{2} + 49 y + (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.2

Expand $(- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5})$ by multiplying each term in the first expression by each term in the second expression.

$y^{2} + 49 y - \frac{y^{2}}{5} \cdot (- \frac{y^{2}}{5}) - \frac{y^{2}}{5} \cdot 10 - \frac{y^{2}}{5} \cdot (- \frac{49 y}{5}) + 10 (- \frac{y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3

Simplify each term.

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Step 2.2.1.1.3.1

Multiply $- \frac{y^{2}}{5} (- \frac{y^{2}}{5})$ .

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Step 2.2.1.1.3.1.1

Multiply $- 1$ by $- 1$ .

$y^{2} + 49 y + 1 (\frac{y^{2}}{5}) \cdot \frac{y^{2}}{5} - \frac{y^{2}}{5} \cdot 10 - \frac{y^{2}}{5} \cdot (- \frac{49 y}{5}) + 10 (- \frac{y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.1.2

Multiply $\frac{y^{2}}{5}$ by $1$ .

$y^{2} + 49 y + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} - \frac{y^{2}}{5} \cdot 10 - \frac{y^{2}}{5} \cdot (- \frac{49 y}{5}) + 10 (- \frac{y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.1.3

Multiply $\frac{y^{2}}{5}$ by $\frac{y^{2}}{5}$ .

$y^{2} + 49 y + \frac{y^{2} y^{2}}{5 \cdot 5} - \frac{y^{2}}{5} \cdot 10 - \frac{y^{2}}{5} \cdot (- \frac{49 y}{5}) + 10 (- \frac{y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.1.4

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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Step 2.2.1.1.3.1.4.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y^{2} + 49 y + \frac{y^{2 + 2}}{5 \cdot 5} - \frac{y^{2}}{5} \cdot 10 - \frac{y^{2}}{5} \cdot (- \frac{49 y}{5}) + 10 (- \frac{y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.1.4.2

Add $2$ and $2$ .

$y^{2} + 49 y + \frac{y^{4}}{5 \cdot 5} - \frac{y^{2}}{5} \cdot 10 - \frac{y^{2}}{5} \cdot (- \frac{49 y}{5}) + 10 (- \frac{y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.1.5

Multiply $5$ by $5$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - \frac{y^{2}}{5} \cdot 10 - \frac{y^{2}}{5} \cdot (- \frac{49 y}{5}) + 10 (- \frac{y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.2

Cancel the common factor of $5$ .

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Step 2.2.1.1.3.2.1

Move the leading negative in $- \frac{y^{2}}{5}$ into the numerator.

$y^{2} + 49 y + \frac{y^{4}}{25} + \frac{- y^{2}}{5} \cdot 10 - \frac{y^{2}}{5} \cdot (- \frac{49 y}{5}) + 10 (- \frac{y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.2.2

Factor $5$ out of $10$ .

$y^{2} + 49 y + \frac{y^{4}}{25} + \frac{- y^{2}}{5} \cdot (5 (2)) - \frac{y^{2}}{5} \cdot (- \frac{49 y}{5}) + 10 (- \frac{y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.2.3

Cancel the common factor.

$y^{2} + 49 y + \frac{y^{4}}{25} + \frac{- y^{2}}{5} \cdot (5 \cdot 2) - \frac{y^{2}}{5} \cdot (- \frac{49 y}{5}) + 10 (- \frac{y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.2.4

Rewrite the expression.

$y^{2} + 49 y + \frac{y^{4}}{25} - y^{2} \cdot 2 - \frac{y^{2}}{5} \cdot (- \frac{49 y}{5}) + 10 (- \frac{y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.3

Multiply $2$ by $- 1$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} - \frac{y^{2}}{5} \cdot (- \frac{49 y}{5}) + 10 (- \frac{y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.4

Multiply $- \frac{y^{2}}{5} (- \frac{49 y}{5})$ .

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Step 2.2.1.1.3.4.1

Multiply $- 1$ by $- 1$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + 1 (\frac{y^{2}}{5}) \cdot \frac{49 y}{5} + 10 (- \frac{y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.4.2

Multiply $\frac{y^{2}}{5}$ by $1$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{y^{2}}{5} \cdot \frac{49 y}{5} + 10 (- \frac{y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.4.3

Multiply $\frac{y^{2}}{5}$ by $\frac{49 y}{5}$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{y^{2} (49 y)}{5 \cdot 5} + 10 (- \frac{y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.4.4

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 2.2.1.1.3.4.4.1

Move $y$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{y \cdot y^{2} \cdot 49}{5 \cdot 5} + 10 (- \frac{y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.4.4.2

Multiply $y$ by $y^{2}$ .

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Step 2.2.1.1.3.4.4.2.1

Raise $y$ to the power of $1$ .

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.4.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{y^{1 + 2} \cdot 49}{5 \cdot 5} + 10 (- \frac{y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.4.4.3

Add $1$ and $2$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{y^{3} \cdot 49}{5 \cdot 5} + 10 (- \frac{y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.4.5

Multiply $5$ by $5$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{y^{3} \cdot 49}{25} + 10 (- \frac{y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.5

Move $49$ to the left of $y^{3}$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 \cdot y^{3}}{25} + 10 (- \frac{y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.6

Cancel the common factor of $5$ .

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Step 2.2.1.1.3.6.1

Move the leading negative in $- \frac{y^{2}}{5}$ into the numerator.

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} + 10 (\frac{- y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.6.2

Factor $5$ out of $10$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} + 5 (2) (\frac{- y^{2}}{5}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.6.3

Cancel the common factor.

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} + 5 \cdot (2 (\frac{- y^{2}}{5})) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.6.4

Rewrite the expression.

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} + 2 (- y^{2}) + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.7

Multiply $- 1$ by $2$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 10 \cdot 10 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.8

Multiply $10$ by $10$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 + 10 (- \frac{49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.9

Cancel the common factor of $5$ .

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Step 2.2.1.1.3.9.1

Move the leading negative in $- \frac{49 y}{5}$ into the numerator.

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 + 10 (\frac{- 49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.9.2

Factor $5$ out of $10$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 + 5 (2) (\frac{- 49 y}{5}) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.9.3

Cancel the common factor.

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 + 5 \cdot (2 (\frac{- 49 y}{5})) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.9.4

Rewrite the expression.

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 + 2 (- 49 y) - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.10

Multiply $- 49$ by $2$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y - \frac{49 y}{5} \cdot (- \frac{y^{2}}{5}) - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.11

Multiply $- \frac{49 y}{5} (- \frac{y^{2}}{5})$ .

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Step 2.2.1.1.3.11.1

Multiply $- 1$ by $- 1$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + 1 (\frac{49 y}{5}) \cdot \frac{y^{2}}{5} - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.11.2

Multiply $\frac{49 y}{5}$ by $1$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + \frac{49 y}{5} \cdot \frac{y^{2}}{5} - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.11.3

Multiply $\frac{49 y}{5}$ by $\frac{y^{2}}{5}$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + \frac{49 y \cdot y^{2}}{5 \cdot 5} - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.11.4

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 2.2.1.1.3.11.4.1

Move $y^{2}$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + \frac{49 (y^{2} y)}{5 \cdot 5} - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.11.4.2

Multiply $y^{2}$ by $y$ .

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Step 2.2.1.1.3.11.4.2.1

Raise $y$ to the power of $1$ .

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.11.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + \frac{49 y^{2 + 1}}{5 \cdot 5} - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.11.4.3

Add $2$ and $1$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + \frac{49 y^{3}}{5 \cdot 5} - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.11.5

Multiply $5$ by $5$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + \frac{49 y^{3}}{25} - \frac{49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.12

Cancel the common factor of $5$ .

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Step 2.2.1.1.3.12.1

Move the leading negative in $- \frac{49 y}{5}$ into the numerator.

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + \frac{49 y^{3}}{25} + \frac{- 49 y}{5} \cdot 10 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.12.2

Factor $5$ out of $10$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + \frac{49 y^{3}}{25} + \frac{- 49 y}{5} \cdot (5 (2)) - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.12.3

Cancel the common factor.

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + \frac{49 y^{3}}{25} + \frac{- 49 y}{5} \cdot (5 \cdot 2) - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.12.4

Rewrite the expression.

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + \frac{49 y^{3}}{25} - 49 y \cdot 2 - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.13

Multiply $2$ by $- 49$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + \frac{49 y^{3}}{25} - 98 y - \frac{49 y}{5} \cdot (- \frac{49 y}{5}) - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.14

Multiply $- \frac{49 y}{5} (- \frac{49 y}{5})$ .

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Step 2.2.1.1.3.14.1

Multiply $- 1$ by $- 1$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + \frac{49 y^{3}}{25} - 98 y + 1 (\frac{49 y}{5}) \cdot \frac{49 y}{5} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.14.2

Multiply $\frac{49 y}{5}$ by $1$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + \frac{49 y^{3}}{25} - 98 y + \frac{49 y}{5} \cdot \frac{49 y}{5} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.14.3

Multiply $\frac{49 y}{5}$ by $\frac{49 y}{5}$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + \frac{49 y^{3}}{25} - 98 y + \frac{49 y (49 y)}{5 \cdot 5} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.14.4

Multiply $49$ by $49$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + \frac{49 y^{3}}{25} - 98 y + \frac{2401 y \cdot y}{5 \cdot 5} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.14.5

Raise $y$ to the power of $1$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + \frac{49 y^{3}}{25} - 98 y + \frac{2401 (y \cdot y)}{5 \cdot 5} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.14.6

Raise $y$ to the power of $1$ .

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.14.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + \frac{49 y^{3}}{25} - 98 y + \frac{2401 y^{1 + 1}}{5 \cdot 5} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.14.8

Add $1$ and $1$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + \frac{49 y^{3}}{25} - 98 y + \frac{2401 y^{2}}{5 \cdot 5} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.3.14.9

Multiply $5$ by $5$ .

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + \frac{49 y^{3}}{25} - 98 y + \frac{2401 y^{2}}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + \frac{49 y^{3}}{25} - 98 y + \frac{2401 y^{2}}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y + \frac{y^{4}}{25} - 2 y^{2} + \frac{49 y^{3}}{25} - 2 y^{2} + 100 - 98 y + \frac{49 y^{3}}{25} - 98 y + \frac{2401 y^{2}}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.4

Combine the numerators over the common denominator.

$y^{2} + 49 y - 2 y^{2} - 2 y^{2} + 100 - 98 y - 98 y + \frac{y^{4} + 49 y^{3} + 49 y^{3} + 2401 y^{2}}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.5

Add $49 y^{3}$ and $49 y^{3}$ .

$y^{2} + 49 y - 2 y^{2} - 2 y^{2} + 100 - 98 y - 98 y + \frac{y^{4} + 98 y^{3} + 2401 y^{2}}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.6

Simplify the numerator.

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Step 2.2.1.1.6.1

Factor $y^{2}$ out of $y^{4} + 98 y^{3} + 2401 y^{2}$ .

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Step 2.2.1.1.6.1.1

Factor $y^{2}$ out of $y^{4}$ .

$y^{2} + 49 y - 2 y^{2} - 2 y^{2} + 100 - 98 y - 98 y + \frac{y^{2} y^{2} + 98 y^{3} + 2401 y^{2}}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.6.1.2

Factor $y^{2}$ out of $98 y^{3}$ .

$y^{2} + 49 y - 2 y^{2} - 2 y^{2} + 100 - 98 y - 98 y + \frac{y^{2} y^{2} + y^{2} (98 y) + 2401 y^{2}}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.6.1.3

Factor $y^{2}$ out of $2401 y^{2}$ .

$y^{2} + 49 y - 2 y^{2} - 2 y^{2} + 100 - 98 y - 98 y + \frac{y^{2} y^{2} + y^{2} (98 y) + y^{2} \cdot 2401}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.6.1.4

Factor $y^{2}$ out of $y^{2} y^{2} + y^{2} (98 y)$ .

$y^{2} + 49 y - 2 y^{2} - 2 y^{2} + 100 - 98 y - 98 y + \frac{y^{2} (y^{2} + 98 y) + y^{2} \cdot 2401}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.6.1.5

Factor $y^{2}$ out of $y^{2} (y^{2} + 98 y) + y^{2} \cdot 2401$ .

$y^{2} + 49 y - 2 y^{2} - 2 y^{2} + 100 - 98 y - 98 y + \frac{y^{2} (y^{2} + 98 y + 2401)}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y - 2 y^{2} - 2 y^{2} + 100 - 98 y - 98 y + \frac{y^{2} (y^{2} + 98 y + 2401)}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.6.2

Factor using the perfect square rule.

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Step 2.2.1.1.6.2.1

Rewrite $2401$ as $49^{2}$ .

$y^{2} + 49 y - 2 y^{2} - 2 y^{2} + 100 - 98 y - 98 y + \frac{y^{2} (y^{2} + 98 y + 49^{2})}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.6.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$98 y = 2 \cdot y \cdot 49$

Step 2.2.1.1.6.2.3

Rewrite the polynomial.

$y^{2} + 49 y - 2 y^{2} - 2 y^{2} + 100 - 98 y - 98 y + \frac{y^{2} (y^{2} + 2 \cdot y \cdot 49 + 49^{2})}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.6.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = y$ and $b = 49$ .

$y^{2} + 49 y - 2 y^{2} - 2 y^{2} + 100 - 98 y - 98 y + \frac{y^{2} {(y + 49)}^{2}}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y - 2 y^{2} - 2 y^{2} + 100 - 98 y - 98 y + \frac{y^{2} {(y + 49)}^{2}}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y - 2 y^{2} - 2 y^{2} + 100 - 98 y - 98 y + \frac{y^{2} {(y + 49)}^{2}}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.7

Subtract $2 y^{2}$ from $- 2 y^{2}$ .

$y^{2} + 49 y - 4 y^{2} + 100 - 98 y - 98 y + \frac{y^{2} {(y + 49)}^{2}}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.8

To write $- 4 y^{2}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$y^{2} + 49 y + 100 - 98 y - 98 y - 4 y^{2} \cdot \frac{25}{25} + \frac{y^{2} {(y + 49)}^{2}}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.9

Combine $- 4 y^{2}$ and $\frac{25}{25}$ .

$y^{2} + 49 y + 100 - 98 y - 98 y + \frac{- 4 y^{2} \cdot 25}{25} + \frac{y^{2} {(y + 49)}^{2}}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.10

Combine the numerators over the common denominator.

$y^{2} + 49 y + 100 - 98 y - 98 y + \frac{- 4 y^{2} \cdot 25 + y^{2} {(y + 49)}^{2}}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.11

Simplify the numerator.

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Step 2.2.1.1.11.1

Factor $y^{2}$ out of $- 4 y^{2} \cdot 25 + y^{2} {(y + 49)}^{2}$ .

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Step 2.2.1.1.11.1.1

Factor $y^{2}$ out of $- 4 y^{2} \cdot 25$ .

$y^{2} + 49 y + 100 - 98 y - 98 y + \frac{y^{2} (- 4 \cdot 25) + y^{2} {(y + 49)}^{2}}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.11.1.2

Factor $y^{2}$ out of $y^{2} {(y + 49)}^{2}$ .

$y^{2} + 49 y + 100 - 98 y - 98 y + \frac{y^{2} (- 4 \cdot 25) + y^{2} ({(y + 49)}^{2})}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.11.1.3

Factor $y^{2}$ out of $y^{2} (- 4 \cdot 25) + y^{2} ({(y + 49)}^{2})$ .

$y^{2} + 49 y + 100 - 98 y - 98 y + \frac{y^{2} (- 4 \cdot 25 + {(y + 49)}^{2})}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y + 100 - 98 y - 98 y + \frac{y^{2} (- 4 \cdot 25 + {(y + 49)}^{2})}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.11.2

Rewrite $4 \cdot 25$ as ${(2 \cdot 5)}^{2}$ .

$y^{2} + 49 y + 100 - 98 y - 98 y + \frac{y^{2} (- {(2 \cdot 5)}^{2} + {(y + 49)}^{2})}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.11.3

Reorder $- {(2 \cdot 5)}^{2}$ and ${(y + 49)}^{2}$ .

$y^{2} + 49 y + 100 - 98 y - 98 y + \frac{y^{2} ({(y + 49)}^{2} - {(2 \cdot 5)}^{2})}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.11.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y + 49$ and $b = 2 \cdot 5$ .

$y^{2} + 49 y + 100 - 98 y - 98 y + \frac{y^{2} ((y + 49 + 2 \cdot 5) (y + 49 - (2 \cdot 5)))}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.11.5

Simplify.

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Step 2.2.1.1.11.5.1

Multiply $2$ by $5$ .

$y^{2} + 49 y + 100 - 98 y - 98 y + \frac{y^{2} ((y + 49 + 10) (y + 49 - (2 \cdot 5)))}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.11.5.2

Add $49$ and $10$ .

$y^{2} + 49 y + 100 - 98 y - 98 y + \frac{y^{2} ((y + 59) (y + 49 - (2 \cdot 5)))}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.11.5.3

Multiply $- (2 \cdot 5)$ .

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Step 2.2.1.1.11.5.3.1

Multiply $2$ by $5$ .

$y^{2} + 49 y + 100 - 98 y - 98 y + \frac{y^{2} ((y + 59) (y + 49 - 1 \cdot 10))}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.11.5.3.2

Multiply $- 1$ by $10$ .

$y^{2} + 49 y + 100 - 98 y - 98 y + \frac{y^{2} ((y + 59) (y + 49 - 10))}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y + 100 - 98 y - 98 y + \frac{y^{2} ((y + 59) (y + 49 - 10))}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.11.5.4

Subtract $10$ from $49$ .

$y^{2} + 49 y + 100 - 98 y - 98 y + \frac{y^{2} ((y + 59) (y + 39))}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y + 100 - 98 y - 98 y + \frac{y^{2} (y + 59) (y + 39)}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y + 100 - 98 y - 98 y + \frac{y^{2} (y + 59) (y + 39)}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.12

To write $100$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$y^{2} + 49 y - 98 y - 98 y + 100 \cdot \frac{25}{25} + \frac{y^{2} (y + 59) (y + 39)}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.13

Combine $100$ and $\frac{25}{25}$ .

$y^{2} + 49 y - 98 y - 98 y + \frac{100 \cdot 25}{25} + \frac{y^{2} (y + 59) (y + 39)}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.14

Combine the numerators over the common denominator.

$y^{2} + 49 y - 98 y - 98 y + \frac{100 \cdot 25 + y^{2} (y + 59) (y + 39)}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.15

Multiply $100$ by $25$ .

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + y^{2} (y + 59) (y + 39)}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.16

Simplify the numerator.

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Step 2.2.1.1.16.1

Apply the distributive property.

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + (y^{2} y + y^{2} \cdot 59) (y + 39)}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.16.2

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 2.2.1.1.16.2.1

Multiply $y^{2}$ by $y$ .

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Step 2.2.1.1.16.2.1.1

Raise $y$ to the power of $1$ .

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + (y^{2} y + y^{2} \cdot 59) (y + 39)}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.16.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + (y^{2 + 1} + y^{2} \cdot 59) (y + 39)}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + (y^{2 + 1} + y^{2} \cdot 59) (y + 39)}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.16.2.2

Add $2$ and $1$ .

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + (y^{3} + y^{2} \cdot 59) (y + 39)}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + (y^{3} + y^{2} \cdot 59) (y + 39)}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.16.3

Move $59$ to the left of $y^{2}$ .

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + (y^{3} + 59 \cdot y^{2}) (y + 39)}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.16.4

Expand $(y^{3} + 59 y^{2}) (y + 39)$ using the FOIL Method.

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Step 2.2.1.1.16.4.1

Apply the distributive property.

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + y^{3} (y + 39) + 59 y^{2} (y + 39)}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.16.4.2

Apply the distributive property.

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + y^{3} y + y^{3} \cdot 39 + 59 y^{2} (y + 39)}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.16.4.3

Apply the distributive property.

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + y^{3} y + y^{3} \cdot 39 + 59 y^{2} y + 59 y^{2} \cdot 39}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + y^{3} y + y^{3} \cdot 39 + 59 y^{2} y + 59 y^{2} \cdot 39}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.16.5

Simplify and combine like terms.

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Step 2.2.1.1.16.5.1

Simplify each term.

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Step 2.2.1.1.16.5.1.1

Multiply $y^{3}$ by $y$ by adding the exponents.

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Step 2.2.1.1.16.5.1.1.1

Multiply $y^{3}$ by $y$ .

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Step 2.2.1.1.16.5.1.1.1.1

Raise $y$ to the power of $1$ .

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + y^{3} y + y^{3} \cdot 39 + 59 y^{2} y + 59 y^{2} \cdot 39}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.16.5.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + y^{3 + 1} + y^{3} \cdot 39 + 59 y^{2} y + 59 y^{2} \cdot 39}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + y^{3 + 1} + y^{3} \cdot 39 + 59 y^{2} y + 59 y^{2} \cdot 39}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.16.5.1.1.2

Add $3$ and $1$ .

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + y^{4} + y^{3} \cdot 39 + 59 y^{2} y + 59 y^{2} \cdot 39}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + y^{4} + y^{3} \cdot 39 + 59 y^{2} y + 59 y^{2} \cdot 39}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.16.5.1.2

Move $39$ to the left of $y^{3}$ .

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + y^{4} + 39 \cdot y^{3} + 59 y^{2} y + 59 y^{2} \cdot 39}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.16.5.1.3

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 2.2.1.1.16.5.1.3.1

Move $y$ .

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + y^{4} + 39 y^{3} + 59 (y \cdot y^{2}) + 59 y^{2} \cdot 39}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.16.5.1.3.2

Multiply $y$ by $y^{2}$ .

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Step 2.2.1.1.16.5.1.3.2.1

Raise $y$ to the power of $1$ .

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + y^{4} + 39 y^{3} + 59 (y \cdot y^{2}) + 59 y^{2} \cdot 39}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.16.5.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + y^{4} + 39 y^{3} + 59 y^{1 + 2} + 59 y^{2} \cdot 39}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + y^{4} + 39 y^{3} + 59 y^{1 + 2} + 59 y^{2} \cdot 39}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.16.5.1.3.3

Add $1$ and $2$ .

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + y^{4} + 39 y^{3} + 59 y^{3} + 59 y^{2} \cdot 39}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + y^{4} + 39 y^{3} + 59 y^{3} + 59 y^{2} \cdot 39}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.16.5.1.4

Multiply $39$ by $59$ .

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + y^{4} + 39 y^{3} + 59 y^{3} + 2301 y^{2}}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + y^{4} + 39 y^{3} + 59 y^{3} + 2301 y^{2}}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.16.5.2

Add $39 y^{3}$ and $59 y^{3}$ .

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + y^{4} + 98 y^{3} + 2301 y^{2}}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y - 98 y - 98 y + \frac{2500 + y^{4} + 98 y^{3} + 2301 y^{2}}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.16.6

Reorder terms.

$y^{2} + 49 y - 98 y - 98 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} + 2500}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y - 98 y - 98 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} + 2500}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.17

Find the common denominator.

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Step 2.2.1.1.17.1

Write $- 98 y$ as a fraction with denominator $1$ .

$y^{2} + 49 y + \frac{- 98 y}{1} - 98 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} + 2500}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.17.2

Multiply $\frac{- 98 y}{1}$ by $\frac{25}{25}$ .

$y^{2} + 49 y + \frac{- 98 y}{1} \cdot \frac{25}{25} - 98 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} + 2500}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.17.3

Multiply $\frac{- 98 y}{1}$ by $\frac{25}{25}$ .

$y^{2} + 49 y + \frac{- 98 y \cdot 25}{25} - 98 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} + 2500}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.17.4

Write $- 98 y$ as a fraction with denominator $1$ .

$y^{2} + 49 y + \frac{- 98 y \cdot 25}{25} + \frac{- 98 y}{1} + \frac{y^{4} + 98 y^{3} + 2301 y^{2} + 2500}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.17.5

Multiply $\frac{- 98 y}{1}$ by $\frac{25}{25}$ .

$y^{2} + 49 y + \frac{- 98 y \cdot 25}{25} + \frac{- 98 y}{1} \cdot \frac{25}{25} + \frac{y^{4} + 98 y^{3} + 2301 y^{2} + 2500}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.17.6

Multiply $\frac{- 98 y}{1}$ by $\frac{25}{25}$ .

$y^{2} + 49 y + \frac{- 98 y \cdot 25}{25} + \frac{- 98 y \cdot 25}{25} + \frac{y^{4} + 98 y^{3} + 2301 y^{2} + 2500}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y + \frac{- 98 y \cdot 25}{25} + \frac{- 98 y \cdot 25}{25} + \frac{y^{4} + 98 y^{3} + 2301 y^{2} + 2500}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.18

Combine the numerators over the common denominator.

$y^{2} + 49 y + \frac{- 98 y \cdot 25 - 98 y \cdot 25 + y^{4} + 98 y^{3} + 2301 y^{2} + 2500}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.19

Simplify each term.

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Step 2.2.1.1.19.1

Multiply $25$ by $- 98$ .

$y^{2} + 49 y + \frac{- 2450 y - 98 y \cdot 25 + y^{4} + 98 y^{3} + 2301 y^{2} + 2500}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.19.2

Multiply $25$ by $- 98$ .

$y^{2} + 49 y + \frac{- 2450 y - 2450 y + y^{4} + 98 y^{3} + 2301 y^{2} + 2500}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y + \frac{- 2450 y - 2450 y + y^{4} + 98 y^{3} + 2301 y^{2} + 2500}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.20

Subtract $2450 y$ from $- 2450 y$ .

$y^{2} + 49 y + \frac{- 4900 y + y^{4} + 98 y^{3} + 2301 y^{2} + 2500}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.21

Reorder terms.

$y^{2} + 49 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} - 4900 y + 2500}{25} - 5 (- \frac{y^{2}}{5} + 10 - \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.22

Apply the distributive property.

$y^{2} + 49 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} - 4900 y + 2500}{25} - 5 (- \frac{y^{2}}{5}) - 5 \cdot 10 - 5 (- \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.23

Simplify.

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Step 2.2.1.1.23.1

Cancel the common factor of $5$ .

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Step 2.2.1.1.23.1.1

Move the leading negative in $- \frac{y^{2}}{5}$ into the numerator.

$y^{2} + 49 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} - 4900 y + 2500}{25} - 5 \frac{- y^{2}}{5} - 5 \cdot 10 - 5 (- \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.23.1.2

Factor $5$ out of $- 5$ .

$y^{2} + 49 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} - 4900 y + 2500}{25} + 5 (- 1) (\frac{- y^{2}}{5}) - 5 \cdot 10 - 5 (- \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.23.1.3

Cancel the common factor.

$y^{2} + 49 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} - 4900 y + 2500}{25} + 5 \cdot (- 1 \frac{- y^{2}}{5}) - 5 \cdot 10 - 5 (- \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.23.1.4

Rewrite the expression.

$y^{2} + 49 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} - 4900 y + 2500}{25} - 1 (- y^{2}) - 5 \cdot 10 - 5 (- \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} - 4900 y + 2500}{25} - 1 (- y^{2}) - 5 \cdot 10 - 5 (- \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.23.2

Multiply $- 1$ by $- 1$ .

$y^{2} + 49 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} - 4900 y + 2500}{25} + 1 y^{2} - 5 \cdot 10 - 5 (- \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.23.3

Multiply $y^{2}$ by $1$ .

$y^{2} + 49 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} - 4900 y + 2500}{25} + y^{2} - 5 \cdot 10 - 5 (- \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.23.4

Multiply $- 5$ by $10$ .

$y^{2} + 49 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} - 4900 y + 2500}{25} + y^{2} - 50 - 5 (- \frac{49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.23.5

Cancel the common factor of $5$ .

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Step 2.2.1.1.23.5.1

Move the leading negative in $- \frac{49 y}{5}$ into the numerator.

$y^{2} + 49 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} - 4900 y + 2500}{25} + y^{2} - 50 - 5 \frac{- 49 y}{5} - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.23.5.2

Factor $5$ out of $- 5$ .

$y^{2} + 49 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} - 4900 y + 2500}{25} + y^{2} - 50 + 5 (- 1) (\frac{- 49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.23.5.3

Cancel the common factor.

$y^{2} + 49 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} - 4900 y + 2500}{25} + y^{2} - 50 + 5 \cdot (- 1 \frac{- 49 y}{5}) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.23.5.4

Rewrite the expression.

$y^{2} + 49 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} - 4900 y + 2500}{25} + y^{2} - 50 - 1 (- 49 y) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} - 4900 y + 2500}{25} + y^{2} - 50 - 1 (- 49 y) - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.1.23.6

Multiply $- 49$ by $- 1$ .

$y^{2} + 49 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} - 4900 y + 2500}{25} + y^{2} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} - 4900 y + 2500}{25} + y^{2} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y^{2} + 49 y + \frac{y^{4} + 98 y^{3} + 2301 y^{2} - 4900 y + 2500}{25} + y^{2} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.2

To write $y^{2}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$49 y + y^{2} \cdot \frac{25}{25} + \frac{y^{4} + 98 y^{3} + 2301 y^{2} - 4900 y + 2500}{25} + y^{2} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.3

Combine $y^{2}$ and $\frac{25}{25}$ .

$49 y + \frac{y^{2} \cdot 25}{25} + \frac{y^{4} + 98 y^{3} + 2301 y^{2} - 4900 y + 2500}{25} + y^{2} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$49 y + \frac{y^{2} \cdot 25 + y^{4} + 98 y^{3} + 2301 y^{2} - 4900 y + 2500}{25} + y^{2} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.5

Add $y^{2} \cdot 25$ and $2301 y^{2}$ .

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Step 2.2.1.5.1

Reorder $y^{2}$ and $25$ .

$49 y + \frac{y^{4} + 98 y^{3} + 25 \cdot y^{2} + 2301 y^{2} - 4900 y + 2500}{25} + y^{2} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.5.2

Add $25 \cdot y^{2}$ and $2301 y^{2}$ .

$49 y + \frac{y^{4} + 98 y^{3} + 2326 \cdot y^{2} - 4900 y + 2500}{25} + y^{2} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$49 y + \frac{y^{4} + 98 y^{3} + 2326 \cdot y^{2} - 4900 y + 2500}{25} + y^{2} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.6

To write $49 y$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$49 y \cdot \frac{25}{25} + \frac{y^{4} + 98 y^{3} + 2326 y^{2} - 4900 y + 2500}{25} + y^{2} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.7

Simplify terms.

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Step 2.2.1.7.1

Combine $49 y$ and $\frac{25}{25}$ .

$\frac{49 y \cdot 25}{25} + \frac{y^{4} + 98 y^{3} + 2326 y^{2} - 4900 y + 2500}{25} + y^{2} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.7.2

Combine the numerators over the common denominator.

$\frac{49 y \cdot 25 + y^{4} + 98 y^{3} + 2326 y^{2} - 4900 y + 2500}{25} + y^{2} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$\frac{49 y \cdot 25 + y^{4} + 98 y^{3} + 2326 y^{2} - 4900 y + 2500}{25} + y^{2} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.8

Simplify the numerator.

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Step 2.2.1.8.1

Multiply $25$ by $49$ .

$\frac{1225 y + y^{4} + 98 y^{3} + 2326 y^{2} - 4900 y + 2500}{25} + y^{2} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.8.2

Subtract $4900 y$ from $1225 y$ .

$\frac{y^{4} + 98 y^{3} + 2326 y^{2} - 3675 y + 2500}{25} + y^{2} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$\frac{y^{4} + 98 y^{3} + 2326 y^{2} - 3675 y + 2500}{25} + y^{2} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.9

To write $y^{2}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$\frac{y^{4} + 98 y^{3} + 2326 y^{2} - 3675 y + 2500}{25} + y^{2} \cdot \frac{25}{25} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.10

Combine $y^{2}$ and $\frac{25}{25}$ .

$\frac{y^{4} + 98 y^{3} + 2326 y^{2} - 3675 y + 2500}{25} + \frac{y^{2} \cdot 25}{25} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.11

Combine the numerators over the common denominator.

$\frac{y^{4} + 98 y^{3} + 2326 y^{2} - 3675 y + 2500 + y^{2} \cdot 25}{25} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.12

Add $2326 y^{2}$ and $y^{2} \cdot 25$ .

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Step 2.2.1.12.1

Reorder $y^{2}$ and $25$ .

$\frac{y^{4} + 98 y^{3} + 2326 y^{2} + 25 \cdot y^{2} - 3675 y + 2500}{25} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.12.2

Add $2326 y^{2}$ and $25 \cdot y^{2}$ .

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 3675 y + 2500}{25} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 3675 y + 2500}{25} - 50 + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.13

To write $- 50$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 3675 y + 2500}{25} - 50 \cdot \frac{25}{25} + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.14

Combine $- 50$ and $\frac{25}{25}$ .

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 3675 y + 2500}{25} + \frac{- 50 \cdot 25}{25} + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.15

Combine the numerators over the common denominator.

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 3675 y + 2500 - 50 \cdot 25}{25} + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.16

Simplify the expression.

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Step 2.2.1.16.1

Multiply $- 50$ by $25$ .

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 3675 y + 2500 - 1250}{25} + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.16.2

Subtract $1250$ from $2500$ .

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 3675 y + 1250}{25} + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 3675 y + 1250}{25} + 49 y - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.17

To write $49 y$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 3675 y + 1250}{25} + 49 y \cdot \frac{25}{25} - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.18

Simplify terms.

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Step 2.2.1.18.1

Combine $49 y$ and $\frac{25}{25}$ .

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 3675 y + 1250}{25} + \frac{49 y \cdot 25}{25} - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.18.2

Combine the numerators over the common denominator.

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 3675 y + 1250 + 49 y \cdot 25}{25} - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 3675 y + 1250 + 49 y \cdot 25}{25} - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.19

Simplify the numerator.

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Step 2.2.1.19.1

Multiply $25$ by $49$ .

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 3675 y + 1250 + 1225 y}{25} - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.19.2

Add $- 3675 y$ and $1225 y$ .

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 2450 y + 1250}{25} - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 2450 y + 1250}{25} - 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.20

To write $- 50$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 2450 y + 1250}{25} - 50 \cdot \frac{25}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.21

Simplify terms.

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Step 2.2.1.21.1

Combine $- 50$ and $\frac{25}{25}$ .

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 2450 y + 1250}{25} + \frac{- 50 \cdot 25}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.21.2

Combine the numerators over the common denominator.

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 2450 y + 1250 - 50 \cdot 25}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 2450 y + 1250 - 50 \cdot 25}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.22

Simplify the numerator.

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Step 2.2.1.22.1

Multiply $- 50$ by $25$ .

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 2450 y + 1250 - 1250}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.22.2

Subtract $1250$ from $1250$ .

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 2450 y + 0}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.22.3

Add $y^{4} + 98 y^{3} + 2351 y^{2} - 2450 y$ and $0$ .

$\frac{y^{4} + 98 y^{3} + 2351 y^{2} - 2450 y}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.22.4

Rewrite $y^{4} + 98 y^{3} + 2351 y^{2} - 2450 y$ in a factored form.

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Step 2.2.1.22.4.1

Factor $y$ out of $y^{4} + 98 y^{3} + 2351 y^{2} - 2450 y$ .

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Step 2.2.1.22.4.1.1

Factor $y$ out of $y^{4}$ .

$\frac{y \cdot y^{3} + 98 y^{3} + 2351 y^{2} - 2450 y}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.22.4.1.2

Factor $y$ out of $98 y^{3}$ .

$\frac{y \cdot y^{3} + y (98 y^{2}) + 2351 y^{2} - 2450 y}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.22.4.1.3

Factor $y$ out of $2351 y^{2}$ .

$\frac{y \cdot y^{3} + y (98 y^{2}) + y (2351 y) - 2450 y}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.22.4.1.4

Factor $y$ out of $- 2450 y$ .

$\frac{y \cdot y^{3} + y (98 y^{2}) + y (2351 y) + y \cdot - 2450}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.22.4.1.5

Factor $y$ out of $y \cdot y^{3} + y (98 y^{2})$ .

$\frac{y (y^{3} + 98 y^{2}) + y (2351 y) + y \cdot - 2450}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.22.4.1.6

Factor $y$ out of $y (y^{3} + 98 y^{2}) + y (2351 y)$ .

$\frac{y (y^{3} + 98 y^{2} + 2351 y) + y \cdot - 2450}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.22.4.1.7

Factor $y$ out of $y (y^{3} + 98 y^{2} + 2351 y) + y \cdot - 2450$ .

$\frac{y (y^{3} + 98 y^{2} + 2351 y - 2450)}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$\frac{y (y^{3} + 98 y^{2} + 2351 y - 2450)}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.22.4.2

Factor $y^{3} + 98 y^{2} + 2351 y - 2450$ using the rational roots test.

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Step 2.2.1.22.4.2.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 2450, \pm 2, \pm 1225, \pm 5, \pm 490, \pm 7, \pm 350, \pm 10, \pm 245, \pm 14, \pm 175, \pm 25, \pm 98, \pm 35, \pm 70, \pm 49, \pm 50$

$q = \pm 1$

Step 2.2.1.22.4.2.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 2450, \pm 2, \pm 1225, \pm 5, \pm 490, \pm 7, \pm 350, \pm 10, \pm 245, \pm 14, \pm 175, \pm 25, \pm 98, \pm 35, \pm 70, \pm 49, \pm 50$

Step 2.2.1.22.4.2.3

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

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Step 2.2.1.22.4.2.3.1

Substitute $1$ into the polynomial.

$1^{3} + 98 \cdot 1^{2} + 2351 \cdot 1 - 2450$

Step 2.2.1.22.4.2.3.2

Raise $1$ to the power of $3$ .

$1 + 98 \cdot 1^{2} + 2351 \cdot 1 - 2450$

Step 2.2.1.22.4.2.3.3

Raise $1$ to the power of $2$ .

$1 + 98 \cdot 1 + 2351 \cdot 1 - 2450$

Step 2.2.1.22.4.2.3.4

Multiply $98$ by $1$ .

$1 + 98 + 2351 \cdot 1 - 2450$

Step 2.2.1.22.4.2.3.5

Add $1$ and $98$ .

$99 + 2351 \cdot 1 - 2450$

Step 2.2.1.22.4.2.3.6

Multiply $2351$ by $1$ .

$99 + 2351 - 2450$

Step 2.2.1.22.4.2.3.7

Add $99$ and $2351$ .

$2450 - 2450$

Step 2.2.1.22.4.2.3.8

Subtract $2450$ from $2450$ .

$0$

Step 2.2.1.22.4.2.4

Since $1$ is a known root, divide the polynomial by $y - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{y^{3} + 98 y^{2} + 2351 y - 2450}{y - 1}$

Step 2.2.1.22.4.2.5

Divide $y^{3} + 98 y^{2} + 2351 y - 2450$ by $y - 1$ .

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Step 2.2.1.22.4.2.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$y$

$1$

$y^{3}$

$98 y^{2}$

$2351 y$

$2450$

Step 2.2.1.22.4.2.5.2

Divide the highest order term in the dividend $y^{3}$ by the highest order term in divisor $y$ .

					$y^{2}$
	$y$	-	$1$		$y^{3}$	+	$98 y^{2}$	+	$2351 y$	-	$2450$

Step 2.2.1.22.4.2.5.3

Multiply the new quotient term by the divisor.

				$y^{2}$
$y$	-	$1$		$y^{3}$	+	$98 y^{2}$	+	$2351 y$	-	$2450$
			+	$y^{3}$	-	$y^{2}$

Step 2.2.1.22.4.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $y^{3} - y^{2}$

				$y^{2}$
$y$	-	$1$		$y^{3}$	+	$98 y^{2}$	+	$2351 y$	-	$2450$
			-	$y^{3}$	+	$y^{2}$

Step 2.2.1.22.4.2.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$y^{2}$
$y$	-	$1$		$y^{3}$	+	$98 y^{2}$	+	$2351 y$	-	$2450$
			-	$y^{3}$	+	$y^{2}$
					+	$99 y^{2}$

Step 2.2.1.22.4.2.5.6

Pull the next terms from the original dividend down into the current dividend.

				$y^{2}$
$y$	-	$1$		$y^{3}$	+	$98 y^{2}$	+	$2351 y$	-	$2450$
			-	$y^{3}$	+	$y^{2}$
					+	$99 y^{2}$	+	$2351 y$

Step 2.2.1.22.4.2.5.7

Divide the highest order term in the dividend $99 y^{2}$ by the highest order term in divisor $y$ .

				$y^{2}$	+	$99 y$
$y$	-	$1$		$y^{3}$	+	$98 y^{2}$	+	$2351 y$	-	$2450$
			-	$y^{3}$	+	$y^{2}$
					+	$99 y^{2}$	+	$2351 y$

Step 2.2.1.22.4.2.5.8

Multiply the new quotient term by the divisor.

				$y^{2}$	+	$99 y$
$y$	-	$1$		$y^{3}$	+	$98 y^{2}$	+	$2351 y$	-	$2450$
			-	$y^{3}$	+	$y^{2}$
					+	$99 y^{2}$	+	$2351 y$
					+	$99 y^{2}$	-	$99 y$

Step 2.2.1.22.4.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $99 y^{2} - 99 y$

				$y^{2}$	+	$99 y$
$y$	-	$1$		$y^{3}$	+	$98 y^{2}$	+	$2351 y$	-	$2450$
			-	$y^{3}$	+	$y^{2}$
					+	$99 y^{2}$	+	$2351 y$
					-	$99 y^{2}$	+	$99 y$

Step 2.2.1.22.4.2.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$y^{2}$	+	$99 y$
$y$	-	$1$		$y^{3}$	+	$98 y^{2}$	+	$2351 y$	-	$2450$
			-	$y^{3}$	+	$y^{2}$
					+	$99 y^{2}$	+	$2351 y$
					-	$99 y^{2}$	+	$99 y$
							+	$2450 y$

Step 2.2.1.22.4.2.5.11

Pull the next terms from the original dividend down into the current dividend.

				$y^{2}$	+	$99 y$
$y$	-	$1$		$y^{3}$	+	$98 y^{2}$	+	$2351 y$	-	$2450$
			-	$y^{3}$	+	$y^{2}$
					+	$99 y^{2}$	+	$2351 y$
					-	$99 y^{2}$	+	$99 y$
							+	$2450 y$	-	$2450$

Step 2.2.1.22.4.2.5.12

Divide the highest order term in the dividend $2450 y$ by the highest order term in divisor $y$ .

				$y^{2}$	+	$99 y$	+	$2450$
$y$	-	$1$		$y^{3}$	+	$98 y^{2}$	+	$2351 y$	-	$2450$
			-	$y^{3}$	+	$y^{2}$
					+	$99 y^{2}$	+	$2351 y$
					-	$99 y^{2}$	+	$99 y$
							+	$2450 y$	-	$2450$

Step 2.2.1.22.4.2.5.13

Multiply the new quotient term by the divisor.

				$y^{2}$	+	$99 y$	+	$2450$
$y$	-	$1$		$y^{3}$	+	$98 y^{2}$	+	$2351 y$	-	$2450$
			-	$y^{3}$	+	$y^{2}$
					+	$99 y^{2}$	+	$2351 y$
					-	$99 y^{2}$	+	$99 y$
							+	$2450 y$	-	$2450$
							+	$2450 y$	-	$2450$

Step 2.2.1.22.4.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $2450 y - 2450$

				$y^{2}$	+	$99 y$	+	$2450$
$y$	-	$1$		$y^{3}$	+	$98 y^{2}$	+	$2351 y$	-	$2450$
			-	$y^{3}$	+	$y^{2}$
					+	$99 y^{2}$	+	$2351 y$
					-	$99 y^{2}$	+	$99 y$
							+	$2450 y$	-	$2450$
							-	$2450 y$	+	$2450$

Step 2.2.1.22.4.2.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$y^{2}$	+	$99 y$	+	$2450$
$y$	-	$1$		$y^{3}$	+	$98 y^{2}$	+	$2351 y$	-	$2450$
			-	$y^{3}$	+	$y^{2}$
					+	$99 y^{2}$	+	$2351 y$
					-	$99 y^{2}$	+	$99 y$
							+	$2450 y$	-	$2450$
							-	$2450 y$	+	$2450$
										$0$

Step 2.2.1.22.4.2.5.16

Since the remander is $0$ , the final answer is the quotient.

$y^{2} + 99 y + 2450$

Step 2.2.1.22.4.2.6

Write $y^{3} + 98 y^{2} + 2351 y - 2450$ as a set of factors.

$\frac{y ((y - 1) (y^{2} + 99 y + 2450))}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$\frac{y ((y - 1) (y^{2} + 99 y + 2450))}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 2.2.1.22.4.3

Factor $y^{2} + 99 y + 2450$ using the AC method.

$\frac{y (y - 1) (y + 49) (y + 50)}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$\frac{y (y - 1) (y + 49) (y + 50)}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$\frac{y (y - 1) (y + 49) (y + 50)}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$\frac{y (y - 1) (y + 49) (y + 50)}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$\frac{y (y - 1) (y + 49) (y + 50)}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$\frac{y (y - 1) (y + 49) (y + 50)}{25} = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 3

Solve for $y$ in $\frac{y (y - 1) (y + 49) (y + 50)}{25} = 0$ .

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Step 3.1

Set the numerator equal to zero.

$y (y - 1) (y + 49) (y + 50) = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 3.2

Solve the equation for $y$ .

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Step 3.2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y = 0$

$y - 1 = 0$

$y + 49 = 0$

$y + 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 3.2.2

Set $y$ equal to $0$ .

$y = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 3.2.3

Set $y - 1$ equal to $0$ and solve for $y$ .

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Step 3.2.3.1

Set $y - 1$ equal to $0$ .

$y - 1 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 3.2.3.2

Add $1$ to both sides of the equation.

$y = 1$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y = 1$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 3.2.4

Set $y + 49$ equal to $0$ and solve for $y$ .

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Step 3.2.4.1

Set $y + 49$ equal to $0$ .

$y + 49 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 3.2.4.2

Subtract $49$ from both sides of the equation.

$y = - 49$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y = - 49$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 3.2.5

Set $y + 50$ equal to $0$ and solve for $y$ .

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Step 3.2.5.1

Set $y + 50$ equal to $0$ .

$y + 50 = 0$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 3.2.5.2

Subtract $50$ from both sides of the equation.

$y = - 50$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y = - 50$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 3.2.6

The final solution is all the values that make $y (y - 1) (y + 49) (y + 50) = 0$ true.

$y = 0, 1, - 49, - 50$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y = 0, 1, - 49, - 50$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

$y = 0, 1, - 49, - 50$

$x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$

Step 4

Replace all occurrences of $y$ with $0$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$ with $0$ .

$x = - \frac{{(0)}^{2}}{5} + 10 - \frac{49 (0)}{5}$

$y = 0$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{{(0)}^{2}}{5} + 10 - \frac{49 (0)}{5}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = 10 + \frac{- {(0)}^{2} - 49 \cdot 0}{5}$

$y = 0$

Step 4.2.1.2

Simplify each term.

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Step 4.2.1.2.1

Raising $0$ to any positive power yields $0$ .

$x = 10 + \frac{- 0 - 49 \cdot 0}{5}$

$y = 0$

Step 4.2.1.2.2

Multiply $- 1$ by $0$ .

$x = 10 + \frac{0 - 49 \cdot 0}{5}$

$y = 0$

Step 4.2.1.2.3

Multiply $- 49$ by $0$ .

$x = 10 + \frac{0 + 0}{5}$

$y = 0$

$x = 10 + \frac{0 + 0}{5}$

$y = 0$

Step 4.2.1.3

Simplify the expression.

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Step 4.2.1.3.1

Add $0$ and $0$ .

$x = 10 + \frac{0}{5}$

$y = 0$

Step 4.2.1.3.2

Divide $0$ by $5$ .

$x = 10 + 0$

$y = 0$

Step 4.2.1.3.3

Add $10$ and $0$ .

$x = 10$

$y = 0$

$x = 10$

$y = 0$

$x = 10$

$y = 0$

$x = 10$

$y = 0$

$x = 10$

$y = 0$

Step 5

Replace all occurrences of $y$ with $1$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$ with $1$ .

$x = - \frac{{(1)}^{2}}{5} + 10 - \frac{49 (1)}{5}$

$y = 1$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- \frac{{(1)}^{2}}{5} + 10 - \frac{49 (1)}{5}$ .

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Step 5.2.1.1

Combine the numerators over the common denominator.

$x = 10 + \frac{- {(1)}^{2} - 49 \cdot 1}{5}$

$y = 1$

Step 5.2.1.2

Simplify each term.

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Step 5.2.1.2.1

One to any power is one.

$x = 10 + \frac{- 1 \cdot 1 - 49 \cdot 1}{5}$

$y = 1$

Step 5.2.1.2.2

Multiply $- 1$ by $1$ .

$x = 10 + \frac{- 1 - 49 \cdot 1}{5}$

$y = 1$

Step 5.2.1.2.3

Multiply $- 49$ by $1$ .

$x = 10 + \frac{- 1 - 49}{5}$

$y = 1$

$x = 10 + \frac{- 1 - 49}{5}$

$y = 1$

Step 5.2.1.3

Simplify the expression.

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Step 5.2.1.3.1

Subtract $49$ from $- 1$ .

$x = 10 + \frac{- 50}{5}$

$y = 1$

Step 5.2.1.3.2

Divide $- 50$ by $5$ .

$x = 10 - 10$

$y = 1$

Step 5.2.1.3.3

Subtract $10$ from $10$ .

$x = 0$

$y = 1$

$x = 0$

$y = 1$

$x = 0$

$y = 1$

$x = 0$

$y = 1$

$x = 0$

$y = 1$

Step 6

Replace all occurrences of $y$ with $0$ in each equation.

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Step 6.1

Replace all occurrences of $y$ in $x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$ with $0$ .

$x = - \frac{{(0)}^{2}}{5} + 10 - \frac{49 (0)}{5}$

$y = 0$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- \frac{{(0)}^{2}}{5} + 10 - \frac{49 (0)}{5}$ .

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Step 6.2.1.1

Combine the numerators over the common denominator.

$x = 10 + \frac{- {(0)}^{2} - 49 \cdot 0}{5}$

$y = 0$

Step 6.2.1.2

Simplify each term.

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Step 6.2.1.2.1

Raising $0$ to any positive power yields $0$ .

$x = 10 + \frac{- 0 - 49 \cdot 0}{5}$

$y = 0$

Step 6.2.1.2.2

Multiply $- 1$ by $0$ .

$x = 10 + \frac{0 - 49 \cdot 0}{5}$

$y = 0$

Step 6.2.1.2.3

Multiply $- 49$ by $0$ .

$x = 10 + \frac{0 + 0}{5}$

$y = 0$

$x = 10 + \frac{0 + 0}{5}$

$y = 0$

Step 6.2.1.3

Simplify the expression.

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Step 6.2.1.3.1

Add $0$ and $0$ .

$x = 10 + \frac{0}{5}$

$y = 0$

Step 6.2.1.3.2

Divide $0$ by $5$ .

$x = 10 + 0$

$y = 0$

Step 6.2.1.3.3

Add $10$ and $0$ .

$x = 10$

$y = 0$

$x = 10$

$y = 0$

$x = 10$

$y = 0$

$x = 10$

$y = 0$

$x = 10$

$y = 0$

Step 7

Replace all occurrences of $y$ with $1$ in each equation.

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Step 7.1

Replace all occurrences of $y$ in $x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$ with $1$ .

$x = - \frac{{(1)}^{2}}{5} + 10 - \frac{49 (1)}{5}$

$y = 1$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $- \frac{{(1)}^{2}}{5} + 10 - \frac{49 (1)}{5}$ .

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Step 7.2.1.1

Combine the numerators over the common denominator.

$x = 10 + \frac{- {(1)}^{2} - 49 \cdot 1}{5}$

$y = 1$

Step 7.2.1.2

Simplify each term.

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Step 7.2.1.2.1

One to any power is one.

$x = 10 + \frac{- 1 \cdot 1 - 49 \cdot 1}{5}$

$y = 1$

Step 7.2.1.2.2

Multiply $- 1$ by $1$ .

$x = 10 + \frac{- 1 - 49 \cdot 1}{5}$

$y = 1$

Step 7.2.1.2.3

Multiply $- 49$ by $1$ .

$x = 10 + \frac{- 1 - 49}{5}$

$y = 1$

$x = 10 + \frac{- 1 - 49}{5}$

$y = 1$

Step 7.2.1.3

Simplify the expression.

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Step 7.2.1.3.1

Subtract $49$ from $- 1$ .

$x = 10 + \frac{- 50}{5}$

$y = 1$

Step 7.2.1.3.2

Divide $- 50$ by $5$ .

$x = 10 - 10$

$y = 1$

Step 7.2.1.3.3

Subtract $10$ from $10$ .

$x = 0$

$y = 1$

$x = 0$

$y = 1$

$x = 0$

$y = 1$

$x = 0$

$y = 1$

$x = 0$

$y = 1$

Step 8

Replace all occurrences of $y$ with $- 49$ in each equation.

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Step 8.1

Replace all occurrences of $y$ in $x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$ with $- 49$ .

$x = - \frac{{(- 49)}^{2}}{5} + 10 - \frac{49 (- 49)}{5}$

$y = - 49$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $- \frac{{(- 49)}^{2}}{5} + 10 - \frac{49 (- 49)}{5}$ .

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Step 8.2.1.1

Combine the numerators over the common denominator.

$x = 10 + \frac{- {(- 49)}^{2} - 49 \cdot - 49}{5}$

$y = - 49$

Step 8.2.1.2

Simplify each term.

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Step 8.2.1.2.1

Raise $- 49$ to the power of $2$ .

$x = 10 + \frac{- 1 \cdot 2401 - 49 \cdot - 49}{5}$

$y = - 49$

Step 8.2.1.2.2

Multiply $- 1$ by $2401$ .

$x = 10 + \frac{- 2401 - 49 \cdot - 49}{5}$

$y = - 49$

Step 8.2.1.2.3

Multiply $- 49$ by $- 49$ .

$x = 10 + \frac{- 2401 + 2401}{5}$

$y = - 49$

$x = 10 + \frac{- 2401 + 2401}{5}$

$y = - 49$

Step 8.2.1.3

Simplify the expression.

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Step 8.2.1.3.1

Add $- 2401$ and $2401$ .

$x = 10 + \frac{0}{5}$

$y = - 49$

Step 8.2.1.3.2

Divide $0$ by $5$ .

$x = 10 + 0$

$y = - 49$

Step 8.2.1.3.3

Add $10$ and $0$ .

$x = 10$

$y = - 49$

$x = 10$

$y = - 49$

$x = 10$

$y = - 49$

$x = 10$

$y = - 49$

$x = 10$

$y = - 49$

Step 9

Replace all occurrences of $y$ with $0$ in each equation.

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Step 9.1

Replace all occurrences of $y$ in $x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$ with $0$ .

$x = - \frac{{(0)}^{2}}{5} + 10 - \frac{49 (0)}{5}$

$y = 0$

Step 9.2

Simplify the right side.

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Step 9.2.1

Simplify $- \frac{{(0)}^{2}}{5} + 10 - \frac{49 (0)}{5}$ .

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Step 9.2.1.1

Combine the numerators over the common denominator.

$x = 10 + \frac{- {(0)}^{2} - 49 \cdot 0}{5}$

$y = 0$

Step 9.2.1.2

Simplify each term.

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Step 9.2.1.2.1

Raising $0$ to any positive power yields $0$ .

$x = 10 + \frac{- 0 - 49 \cdot 0}{5}$

$y = 0$

Step 9.2.1.2.2

Multiply $- 1$ by $0$ .

$x = 10 + \frac{0 - 49 \cdot 0}{5}$

$y = 0$

Step 9.2.1.2.3

Multiply $- 49$ by $0$ .

$x = 10 + \frac{0 + 0}{5}$

$y = 0$

$x = 10 + \frac{0 + 0}{5}$

$y = 0$

Step 9.2.1.3

Simplify the expression.

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Step 9.2.1.3.1

Add $0$ and $0$ .

$x = 10 + \frac{0}{5}$

$y = 0$

Step 9.2.1.3.2

Divide $0$ by $5$ .

$x = 10 + 0$

$y = 0$

Step 9.2.1.3.3

Add $10$ and $0$ .

$x = 10$

$y = 0$

$x = 10$

$y = 0$

$x = 10$

$y = 0$

$x = 10$

$y = 0$

$x = 10$

$y = 0$

Step 10

Replace all occurrences of $y$ with $1$ in each equation.

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Step 10.1

Replace all occurrences of $y$ in $x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$ with $1$ .

$x = - \frac{{(1)}^{2}}{5} + 10 - \frac{49 (1)}{5}$

$y = 1$

Step 10.2

Simplify the right side.

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Step 10.2.1

Simplify $- \frac{{(1)}^{2}}{5} + 10 - \frac{49 (1)}{5}$ .

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Step 10.2.1.1

Combine the numerators over the common denominator.

$x = 10 + \frac{- {(1)}^{2} - 49 \cdot 1}{5}$

$y = 1$

Step 10.2.1.2

Simplify each term.

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Step 10.2.1.2.1

One to any power is one.

$x = 10 + \frac{- 1 \cdot 1 - 49 \cdot 1}{5}$

$y = 1$

Step 10.2.1.2.2

Multiply $- 1$ by $1$ .

$x = 10 + \frac{- 1 - 49 \cdot 1}{5}$

$y = 1$

Step 10.2.1.2.3

Multiply $- 49$ by $1$ .

$x = 10 + \frac{- 1 - 49}{5}$

$y = 1$

$x = 10 + \frac{- 1 - 49}{5}$

$y = 1$

Step 10.2.1.3

Simplify the expression.

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Step 10.2.1.3.1

Subtract $49$ from $- 1$ .

$x = 10 + \frac{- 50}{5}$

$y = 1$

Step 10.2.1.3.2

Divide $- 50$ by $5$ .

$x = 10 - 10$

$y = 1$

Step 10.2.1.3.3

Subtract $10$ from $10$ .

$x = 0$

$y = 1$

$x = 0$

$y = 1$

$x = 0$

$y = 1$

$x = 0$

$y = 1$

$x = 0$

$y = 1$

Step 11

Replace all occurrences of $y$ with $- 49$ in each equation.

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Step 11.1

Replace all occurrences of $y$ in $x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$ with $- 49$ .

$x = - \frac{{(- 49)}^{2}}{5} + 10 - \frac{49 (- 49)}{5}$

$y = - 49$

Step 11.2

Simplify the right side.

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Step 11.2.1

Simplify $- \frac{{(- 49)}^{2}}{5} + 10 - \frac{49 (- 49)}{5}$ .

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Step 11.2.1.1

Combine the numerators over the common denominator.

$x = 10 + \frac{- {(- 49)}^{2} - 49 \cdot - 49}{5}$

$y = - 49$

Step 11.2.1.2

Simplify each term.

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Step 11.2.1.2.1

Raise $- 49$ to the power of $2$ .

$x = 10 + \frac{- 1 \cdot 2401 - 49 \cdot - 49}{5}$

$y = - 49$

Step 11.2.1.2.2

Multiply $- 1$ by $2401$ .

$x = 10 + \frac{- 2401 - 49 \cdot - 49}{5}$

$y = - 49$

Step 11.2.1.2.3

Multiply $- 49$ by $- 49$ .

$x = 10 + \frac{- 2401 + 2401}{5}$

$y = - 49$

$x = 10 + \frac{- 2401 + 2401}{5}$

$y = - 49$

Step 11.2.1.3

Simplify the expression.

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Step 11.2.1.3.1

Add $- 2401$ and $2401$ .

$x = 10 + \frac{0}{5}$

$y = - 49$

Step 11.2.1.3.2

Divide $0$ by $5$ .

$x = 10 + 0$

$y = - 49$

Step 11.2.1.3.3

Add $10$ and $0$ .

$x = 10$

$y = - 49$

$x = 10$

$y = - 49$

$x = 10$

$y = - 49$

$x = 10$

$y = - 49$

$x = 10$

$y = - 49$

Step 12

Replace all occurrences of $y$ with $- 50$ in each equation.

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Step 12.1

Replace all occurrences of $y$ in $x = - \frac{y^{2}}{5} + 10 - \frac{49 y}{5}$ with $- 50$ .

$x = - \frac{{(- 50)}^{2}}{5} + 10 - \frac{49 (- 50)}{5}$

$y = - 50$

Step 12.2

Simplify the right side.

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Step 12.2.1

Simplify $- \frac{{(- 50)}^{2}}{5} + 10 - \frac{49 (- 50)}{5}$ .

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Step 12.2.1.1

Combine the numerators over the common denominator.

$x = 10 + \frac{- {(- 50)}^{2} - 49 \cdot - 50}{5}$

$y = - 50$

Step 12.2.1.2

Simplify each term.

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Step 12.2.1.2.1

Raise $- 50$ to the power of $2$ .

$x = 10 + \frac{- 1 \cdot 2500 - 49 \cdot - 50}{5}$

$y = - 50$

Step 12.2.1.2.2

Multiply $- 1$ by $2500$ .

$x = 10 + \frac{- 2500 - 49 \cdot - 50}{5}$

$y = - 50$

Step 12.2.1.2.3

Multiply $- 49$ by $- 50$ .

$x = 10 + \frac{- 2500 + 2450}{5}$

$y = - 50$

$x = 10 + \frac{- 2500 + 2450}{5}$

$y = - 50$

Step 12.2.1.3

Simplify the expression.

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Step 12.2.1.3.1

Add $- 2500$ and $2450$ .

$x = 10 + \frac{- 50}{5}$

$y = - 50$

Step 12.2.1.3.2

Divide $- 50$ by $5$ .

$x = 10 - 10$

$y = - 50$

Step 12.2.1.3.3

Subtract $10$ from $10$ .

$x = 0$

$y = - 50$

$x = 0$

$y = - 50$

$x = 0$

$y = - 50$

$x = 0$

$y = - 50$

$x = 0$

$y = - 50$

Step 13

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(10, 0)$

$(0, 1)$

$(10, - 49)$

$(0, - 50)$

Step 14

The result can be shown in multiple forms.

Point Form:

$(10, 0), (0, 1), (10, - 49), (0, - 50)$

Equation Form:

$x = 10, y = 0$

$x = 0, y = 1$

$x = 10, y = - 49$

$x = 0, y = - 50$

Step 15