Precalculus Examples

$4 x^{2} - 3 x y + 9 y^{2} = 15$ , $2 x + 3 y = 5$

Step 1

Solve for $x$ in $2 x + 3 y = 5$ .

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Step 1.1

Subtract $3 y$ from both sides of the equation.

$2 x = 5 - 3 y$

$4 x^{2} - 3 x y + 9 y^{2} = 15$

Step 1.2

Divide each term in $2 x = 5 - 3 y$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 x = 5 - 3 y$ by $2$ .

$\frac{2 x}{2} = \frac{5}{2} + \frac{- 3 y}{2}$

$4 x^{2} - 3 x y + 9 y^{2} = 15$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{5}{2} + \frac{- 3 y}{2}$

$4 x^{2} - 3 x y + 9 y^{2} = 15$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{2} + \frac{- 3 y}{2}$

$4 x^{2} - 3 x y + 9 y^{2} = 15$

$x = \frac{5}{2} + \frac{- 3 y}{2}$

$4 x^{2} - 3 x y + 9 y^{2} = 15$

$x = \frac{5}{2} + \frac{- 3 y}{2}$

$4 x^{2} - 3 x y + 9 y^{2} = 15$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$x = \frac{5}{2} - \frac{3 y}{2}$

$4 x^{2} - 3 x y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$4 x^{2} - 3 x y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$4 x^{2} - 3 x y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$4 x^{2} - 3 x y + 9 y^{2} = 15$

Step 2

Replace all occurrences of $x$ with $\frac{5}{2} - \frac{3 y}{2}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $4 x^{2} - 3 x y + 9 y^{2} = 15$ with $\frac{5}{2} - \frac{3 y}{2}$ .

$4 {(\frac{5}{2} - \frac{3 y}{2})}^{2} - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $4 {(\frac{5}{2} - \frac{3 y}{2})}^{2} - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(\frac{5}{2} - \frac{3 y}{2})}^{2}$ as $(\frac{5}{2} - \frac{3 y}{2}) (\frac{5}{2} - \frac{3 y}{2})$ .

$4 ((\frac{5}{2} - \frac{3 y}{2}) (\frac{5}{2} - \frac{3 y}{2})) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.2

Expand $(\frac{5}{2} - \frac{3 y}{2}) (\frac{5}{2} - \frac{3 y}{2})$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$4 (\frac{5}{2} \cdot (\frac{5}{2} - \frac{3 y}{2}) - \frac{3 y}{2} \cdot (\frac{5}{2} - \frac{3 y}{2})) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.2.2

Apply the distributive property.

$4 (\frac{5}{2} \cdot \frac{5}{2} + \frac{5}{2} \cdot (- \frac{3 y}{2}) - \frac{3 y}{2} \cdot (\frac{5}{2} - \frac{3 y}{2})) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.2.3

Apply the distributive property.

$4 (\frac{5}{2} \cdot \frac{5}{2} + \frac{5}{2} \cdot (- \frac{3 y}{2}) - \frac{3 y}{2} \cdot \frac{5}{2} - \frac{3 y}{2} \cdot (- \frac{3 y}{2})) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.3

Simplify and combine like terms.

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Step 2.2.1.1.3.1

Simplify each term.

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Step 2.2.1.1.3.1.1

Multiply $\frac{5}{2} \cdot \frac{5}{2}$ .

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Step 2.2.1.1.3.1.1.1

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

$4 (\frac{5 \cdot 5}{2 \cdot 2} + \frac{5}{2} \cdot (- \frac{3 y}{2}) - \frac{3 y}{2} \cdot \frac{5}{2} - \frac{3 y}{2} \cdot (- \frac{3 y}{2})) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.3.1.1.2

Multiply $5$ by $5$ .

$4 (\frac{25}{2 \cdot 2} + \frac{5}{2} \cdot (- \frac{3 y}{2}) - \frac{3 y}{2} \cdot \frac{5}{2} - \frac{3 y}{2} \cdot (- \frac{3 y}{2})) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.3.1.1.3

Multiply $2$ by $2$ .

$4 (\frac{25}{4} + \frac{5}{2} \cdot (- \frac{3 y}{2}) - \frac{3 y}{2} \cdot \frac{5}{2} - \frac{3 y}{2} \cdot (- \frac{3 y}{2})) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$4 (\frac{25}{4} + \frac{5}{2} \cdot (- \frac{3 y}{2}) - \frac{3 y}{2} \cdot \frac{5}{2} - \frac{3 y}{2} \cdot (- \frac{3 y}{2})) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.3.1.2

Multiply $\frac{5}{2} (- \frac{3 y}{2})$ .

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Step 2.2.1.1.3.1.2.1

Multiply $\frac{5}{2}$ by $\frac{3 y}{2}$ .

$4 (\frac{25}{4} - \frac{5 (3 y)}{2 \cdot 2} - \frac{3 y}{2} \cdot \frac{5}{2} - \frac{3 y}{2} \cdot (- \frac{3 y}{2})) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.3.1.2.2

Multiply $3$ by $5$ .

$4 (\frac{25}{4} - \frac{15 y}{2 \cdot 2} - \frac{3 y}{2} \cdot \frac{5}{2} - \frac{3 y}{2} \cdot (- \frac{3 y}{2})) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.3.1.2.3

Multiply $2$ by $2$ .

$4 (\frac{25}{4} - \frac{15 y}{4} - \frac{3 y}{2} \cdot \frac{5}{2} - \frac{3 y}{2} \cdot (- \frac{3 y}{2})) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$4 (\frac{25}{4} - \frac{15 y}{4} - \frac{3 y}{2} \cdot \frac{5}{2} - \frac{3 y}{2} \cdot (- \frac{3 y}{2})) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.3.1.3

Multiply $- \frac{3 y}{2} \cdot \frac{5}{2}$ .

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Step 2.2.1.1.3.1.3.1

Multiply $\frac{5}{2}$ by $\frac{3 y}{2}$ .

$4 (\frac{25}{4} - \frac{15 y}{4} - \frac{5 (3 y)}{2 \cdot 2} - \frac{3 y}{2} \cdot (- \frac{3 y}{2})) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.3.1.3.2

Multiply $3$ by $5$ .

$4 (\frac{25}{4} - \frac{15 y}{4} - \frac{15 y}{2 \cdot 2} - \frac{3 y}{2} \cdot (- \frac{3 y}{2})) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.3.1.3.3

Multiply $2$ by $2$ .

$4 (\frac{25}{4} - \frac{15 y}{4} - \frac{15 y}{4} - \frac{3 y}{2} \cdot (- \frac{3 y}{2})) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$4 (\frac{25}{4} - \frac{15 y}{4} - \frac{15 y}{4} - \frac{3 y}{2} \cdot (- \frac{3 y}{2})) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.3.1.4

Multiply $- \frac{3 y}{2} (- \frac{3 y}{2})$ .

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Step 2.2.1.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$4 (\frac{25}{4} - \frac{15 y}{4} - \frac{15 y}{4} + 1 (\frac{3 y}{2}) \cdot \frac{3 y}{2}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.3.1.4.2

Multiply $\frac{3 y}{2}$ by $1$ .

$4 (\frac{25}{4} - \frac{15 y}{4} - \frac{15 y}{4} + \frac{3 y}{2} \cdot \frac{3 y}{2}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.3.1.4.3

Multiply $\frac{3 y}{2}$ by $\frac{3 y}{2}$ .

$4 (\frac{25}{4} - \frac{15 y}{4} - \frac{15 y}{4} + \frac{3 y (3 y)}{2 \cdot 2}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.3.1.4.4

Multiply $3$ by $3$ .

$4 (\frac{25}{4} - \frac{15 y}{4} - \frac{15 y}{4} + \frac{9 y \cdot y}{2 \cdot 2}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.3.1.4.5

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

$4 (\frac{25}{4} - \frac{15 y}{4} - \frac{15 y}{4} + \frac{9 (y \cdot y)}{2 \cdot 2}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.3.1.4.6

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

$4 (\frac{25}{4} - \frac{15 y}{4} - \frac{15 y}{4} + \frac{9 (y \cdot y)}{2 \cdot 2}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.3.1.4.7

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

$4 (\frac{25}{4} - \frac{15 y}{4} - \frac{15 y}{4} + \frac{9 y^{1 + 1}}{2 \cdot 2}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.3.1.4.8

Add $1$ and $1$ .

$4 (\frac{25}{4} - \frac{15 y}{4} - \frac{15 y}{4} + \frac{9 y^{2}}{2 \cdot 2}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.3.1.4.9

Multiply $2$ by $2$ .

$4 (\frac{25}{4} - \frac{15 y}{4} - \frac{15 y}{4} + \frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$4 (\frac{25}{4} - \frac{15 y}{4} - \frac{15 y}{4} + \frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$4 (\frac{25}{4} - \frac{15 y}{4} - \frac{15 y}{4} + \frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.3.2

Subtract $\frac{15 y}{4}$ from $- \frac{15 y}{4}$ .

$4 (\frac{25}{4} - 2 \frac{15 y}{4} + \frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$4 (\frac{25}{4} - 2 \frac{15 y}{4} + \frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.4

Simplify each term.

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Step 2.2.1.1.4.1

Cancel the common factor of $2$ .

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Step 2.2.1.1.4.1.1

Factor $2$ out of $- 2$ .

$4 (\frac{25}{4} + 2 (- 1) (\frac{15 y}{4}) + \frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.4.1.2

Factor $2$ out of $4$ .

$4 (\frac{25}{4} + 2 \cdot (- 1 \frac{15 y}{2 \cdot 2}) + \frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.4.1.3

Cancel the common factor.

$4 (\frac{25}{4} + 2 \cdot (- 1 \frac{15 y}{2 \cdot 2}) + \frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.4.1.4

Rewrite the expression.

$4 (\frac{25}{4} - 1 \frac{15 y}{2} + \frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$4 (\frac{25}{4} - 1 \frac{15 y}{2} + \frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.4.2

Rewrite $- 1 \frac{15 y}{2}$ as $- \frac{15 y}{2}$ .

$4 (\frac{25}{4} - \frac{15 y}{2} + \frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$4 (\frac{25}{4} - \frac{15 y}{2} + \frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.5

Apply the distributive property.

$4 (\frac{25}{4}) + 4 (- \frac{15 y}{2}) + 4 (\frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.6

Simplify.

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Step 2.2.1.1.6.1

Cancel the common factor of $4$ .

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Step 2.2.1.1.6.1.1

Cancel the common factor.

$4 (\frac{25}{4}) + 4 (- \frac{15 y}{2}) + 4 (\frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.6.1.2

Rewrite the expression.

$25 + 4 (- \frac{15 y}{2}) + 4 (\frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$25 + 4 (- \frac{15 y}{2}) + 4 (\frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.6.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.6.2.1

Move the leading negative in $- \frac{15 y}{2}$ into the numerator.

$25 + 4 (\frac{- 15 y}{2}) + 4 (\frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.6.2.2

Factor $2$ out of $4$ .

$25 + 2 (2) (\frac{- 15 y}{2}) + 4 (\frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.6.2.3

Cancel the common factor.

$25 + 2 \cdot (2 (\frac{- 15 y}{2})) + 4 (\frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.6.2.4

Rewrite the expression.

$25 + 2 (- 15 y) + 4 (\frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$25 + 2 (- 15 y) + 4 (\frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.6.3

Multiply $- 15$ by $2$ .

$25 - 30 y + 4 (\frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.6.4

Cancel the common factor of $4$ .

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Step 2.2.1.1.6.4.1

Cancel the common factor.

$25 - 30 y + 4 (\frac{9 y^{2}}{4}) - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.6.4.2

Rewrite the expression.

$25 - 30 y + 9 y^{2} - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$25 - 30 y + 9 y^{2} - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$25 - 30 y + 9 y^{2} - 3 (\frac{5}{2} - \frac{3 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.7

Apply the distributive property.

$25 - 30 y + 9 y^{2} + (- 3 (\frac{5}{2}) - 3 (- \frac{3 y}{2})) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.8

Multiply $- 3 (\frac{5}{2})$ .

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Step 2.2.1.1.8.1

Combine $- 3$ and $\frac{5}{2}$ .

$25 - 30 y + 9 y^{2} + (\frac{- 3 \cdot 5}{2} - 3 (- \frac{3 y}{2})) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.8.2

Multiply $- 3$ by $5$ .

$25 - 30 y + 9 y^{2} + (\frac{- 15}{2} - 3 (- \frac{3 y}{2})) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$25 - 30 y + 9 y^{2} + (\frac{- 15}{2} - 3 (- \frac{3 y}{2})) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.9

Multiply $- 3 (- \frac{3 y}{2})$ .

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Step 2.2.1.1.9.1

Multiply $- 1$ by $- 3$ .

$25 - 30 y + 9 y^{2} + (\frac{- 15}{2} + 3 (\frac{3 y}{2})) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.9.2

Combine $3$ and $\frac{3 y}{2}$ .

$25 - 30 y + 9 y^{2} + (\frac{- 15}{2} + \frac{3 (3 y)}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.9.3

Multiply $3$ by $3$ .

$25 - 30 y + 9 y^{2} + (\frac{- 15}{2} + \frac{9 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$25 - 30 y + 9 y^{2} + (\frac{- 15}{2} + \frac{9 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.10

Move the negative in front of the fraction.

$25 - 30 y + 9 y^{2} + (- \frac{15}{2} + \frac{9 y}{2}) \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.11

Apply the distributive property.

$25 - 30 y + 9 y^{2} - \frac{15}{2} y + \frac{9 y}{2} \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.12

Combine $y$ and $\frac{15}{2}$ .

$25 - 30 y + 9 y^{2} - \frac{y \cdot 15}{2} + \frac{9 y}{2} \cdot y + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.13

Multiply $\frac{9 y}{2} y$ .

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Step 2.2.1.1.13.1

Combine $\frac{9 y}{2}$ and $y$ .

$25 - 30 y + 9 y^{2} - \frac{y \cdot 15}{2} + \frac{9 y \cdot y}{2} + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.13.2

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

$25 - 30 y + 9 y^{2} - \frac{y \cdot 15}{2} + \frac{9 (y \cdot y)}{2} + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.13.3

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

$25 - 30 y + 9 y^{2} - \frac{y \cdot 15}{2} + \frac{9 (y \cdot y)}{2} + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.13.4

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

$25 - 30 y + 9 y^{2} - \frac{y \cdot 15}{2} + \frac{9 y^{1 + 1}}{2} + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.13.5

Add $1$ and $1$ .

$25 - 30 y + 9 y^{2} - \frac{y \cdot 15}{2} + \frac{9 y^{2}}{2} + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$25 - 30 y + 9 y^{2} - \frac{y \cdot 15}{2} + \frac{9 y^{2}}{2} + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.1.14

Move $15$ to the left of $y$ .

$25 - 30 y + 9 y^{2} - \frac{15 y}{2} + \frac{9 y^{2}}{2} + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$25 - 30 y + 9 y^{2} - \frac{15 y}{2} + \frac{9 y^{2}}{2} + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.2

To write $- 30 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$25 + 9 y^{2} - 30 y \cdot \frac{2}{2} - \frac{15 y}{2} + \frac{9 y^{2}}{2} + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.3

Combine $- 30 y$ and $\frac{2}{2}$ .

$25 + 9 y^{2} + \frac{- 30 y \cdot 2}{2} - \frac{15 y}{2} + \frac{9 y^{2}}{2} + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$25 + 9 y^{2} + \frac{- 30 y \cdot 2 - 15 y}{2} + \frac{9 y^{2}}{2} + 9 y^{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$25 + 9 y^{2} + 9 y^{2} + \frac{- 30 y \cdot 2 - 15 y + 9 y^{2}}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.6

Multiply $2$ by $- 30$ .

$25 + 9 y^{2} + 9 y^{2} + \frac{- 60 y - 15 y + 9 y^{2}}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.7

Subtract $15 y$ from $- 60 y$ .

$25 + 9 y^{2} + 9 y^{2} + \frac{- 75 y + 9 y^{2}}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.8

Factor $3 y$ out of $- 75 y + 9 y^{2}$ .

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Step 2.2.1.8.1

Factor $3 y$ out of $- 75 y$ .

$25 + 9 y^{2} + 9 y^{2} + \frac{3 y (- 25) + 9 y^{2}}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.8.2

Factor $3 y$ out of $9 y^{2}$ .

$25 + 9 y^{2} + 9 y^{2} + \frac{3 y (- 25) + 3 y (3 y)}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.8.3

Factor $3 y$ out of $3 y (- 25) + 3 y (3 y)$ .

$25 + 9 y^{2} + 9 y^{2} + \frac{3 y (- 25 + 3 y)}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$25 + 9 y^{2} + 9 y^{2} + \frac{3 y (- 25 + 3 y)}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.9

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

$9 y^{2} + 9 y^{2} + 25 \cdot \frac{2}{2} + \frac{3 y (- 25 + 3 y)}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.10

Simplify terms.

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Step 2.2.1.10.1

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

$9 y^{2} + 9 y^{2} + \frac{25 \cdot 2}{2} + \frac{3 y (- 25 + 3 y)}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.10.2

Combine the numerators over the common denominator.

$9 y^{2} + 9 y^{2} + \frac{25 \cdot 2 + 3 y (- 25 + 3 y)}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.10.3

Multiply $25$ by $2$ .

$9 y^{2} + 9 y^{2} + \frac{50 + 3 y (- 25 + 3 y)}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$9 y^{2} + 9 y^{2} + \frac{50 + 3 y (- 25 + 3 y)}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.11

Simplify the numerator.

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Step 2.2.1.11.1

Apply the distributive property.

$9 y^{2} + 9 y^{2} + \frac{50 + 3 y \cdot - 25 + 3 y (3 y)}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.11.2

Multiply $- 25$ by $3$ .

$9 y^{2} + 9 y^{2} + \frac{50 - 75 y + 3 y (3 y)}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.11.3

Rewrite using the commutative property of multiplication.

$9 y^{2} + 9 y^{2} + \frac{50 - 75 y + 3 \cdot (3 y \cdot y)}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.11.4

Simplify each term.

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Step 2.2.1.11.4.1

Multiply $y$ by $y$ by adding the exponents.

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Step 2.2.1.11.4.1.1

Move $y$ .

$9 y^{2} + 9 y^{2} + \frac{50 - 75 y + 3 \cdot (3 (y \cdot y))}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.11.4.1.2

Multiply $y$ by $y$ .

$9 y^{2} + 9 y^{2} + \frac{50 - 75 y + 3 \cdot (3 y^{2})}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$9 y^{2} + 9 y^{2} + \frac{50 - 75 y + 3 \cdot (3 y^{2})}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.11.4.2

Multiply $3$ by $3$ .

$9 y^{2} + 9 y^{2} + \frac{50 - 75 y + 9 y^{2}}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$9 y^{2} + 9 y^{2} + \frac{50 - 75 y + 9 y^{2}}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$9 y^{2} + 9 y^{2} + \frac{50 - 75 y + 9 y^{2}}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.12

Find the common denominator.

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Step 2.2.1.12.1

Write $9 y^{2}$ as a fraction with denominator $1$ .

$\frac{9 y^{2}}{1} + 9 y^{2} + \frac{50 - 75 y + 9 y^{2}}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.12.2

Multiply $\frac{9 y^{2}}{1}$ by $\frac{2}{2}$ .

$\frac{9 y^{2}}{1} \cdot \frac{2}{2} + 9 y^{2} + \frac{50 - 75 y + 9 y^{2}}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.12.3

Multiply $\frac{9 y^{2}}{1}$ by $\frac{2}{2}$ .

$\frac{9 y^{2} \cdot 2}{2} + 9 y^{2} + \frac{50 - 75 y + 9 y^{2}}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.12.4

Write $9 y^{2}$ as a fraction with denominator $1$ .

$\frac{9 y^{2} \cdot 2}{2} + \frac{9 y^{2}}{1} + \frac{50 - 75 y + 9 y^{2}}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.12.5

Multiply $\frac{9 y^{2}}{1}$ by $\frac{2}{2}$ .

$\frac{9 y^{2} \cdot 2}{2} + \frac{9 y^{2}}{1} \cdot \frac{2}{2} + \frac{50 - 75 y + 9 y^{2}}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.12.6

Multiply $\frac{9 y^{2}}{1}$ by $\frac{2}{2}$ .

$\frac{9 y^{2} \cdot 2}{2} + \frac{9 y^{2} \cdot 2}{2} + \frac{50 - 75 y + 9 y^{2}}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$\frac{9 y^{2} \cdot 2}{2} + \frac{9 y^{2} \cdot 2}{2} + \frac{50 - 75 y + 9 y^{2}}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.13

Simplify terms.

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Step 2.2.1.13.1

Combine the numerators over the common denominator.

$\frac{9 y^{2} \cdot 2 + 9 y^{2} \cdot 2 + 50 - 75 y + 9 y^{2}}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.13.2

Simplify each term.

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Step 2.2.1.13.2.1

Multiply $2$ by $9$ .

$\frac{18 y^{2} + 9 y^{2} \cdot 2 + 50 - 75 y + 9 y^{2}}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.13.2.2

Multiply $2$ by $9$ .

$\frac{18 y^{2} + 18 y^{2} + 50 - 75 y + 9 y^{2}}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$\frac{18 y^{2} + 18 y^{2} + 50 - 75 y + 9 y^{2}}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.13.3

Simplify by adding terms.

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Step 2.2.1.13.3.1

Add $18 y^{2}$ and $18 y^{2}$ .

$\frac{36 y^{2} + 50 - 75 y + 9 y^{2}}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.13.3.2

Add $36 y^{2}$ and $9 y^{2}$ .

$\frac{45 y^{2} + 50 - 75 y}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$\frac{45 y^{2} + 50 - 75 y}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$\frac{45 y^{2} + 50 - 75 y}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.14

Simplify the numerator.

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Step 2.2.1.14.1

Factor $5$ out of $45 y^{2} + 50 - 75 y$ .

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Step 2.2.1.14.1.1

Factor $5$ out of $45 y^{2}$ .

$\frac{5 (9 y^{2}) + 50 - 75 y}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.14.1.2

Factor $5$ out of $50$ .

$\frac{5 (9 y^{2}) + 5 (10) - 75 y}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.14.1.3

Factor $5$ out of $- 75 y$ .

$\frac{5 (9 y^{2}) + 5 (10) + 5 (- 15 y)}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.14.1.4

Factor $5$ out of $5 (9 y^{2}) + 5 (10)$ .

$\frac{5 (9 y^{2} + 10) + 5 (- 15 y)}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.14.1.5

Factor $5$ out of $5 (9 y^{2} + 10) + 5 (- 15 y)$ .

$\frac{5 (9 y^{2} + 10 - 15 y)}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$\frac{5 (9 y^{2} + 10 - 15 y)}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 2.2.1.14.2

Reorder terms.

$\frac{5 (9 y^{2} - 15 y + 10)}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$\frac{5 (9 y^{2} - 15 y + 10)}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$\frac{5 (9 y^{2} - 15 y + 10)}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$\frac{5 (9 y^{2} - 15 y + 10)}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

$\frac{5 (9 y^{2} - 15 y + 10)}{2} = 15$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3

Solve for $y$ in $\frac{5 (9 y^{2} - 15 y + 10)}{2} = 15$ .

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Step 3.1

Multiply both sides by $2$ .

$\frac{5 (9 y^{2} - 15 y + 10)}{2} \cdot 2 = 15 \cdot 2$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $\frac{5 (9 y^{2} - 15 y + 10)}{2} \cdot 2$ .

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Step 3.2.1.1.1

Cancel the common factor of $2$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{5 (9 y^{2} - 15 y + 10)}{2} \cdot 2 = 15 \cdot 2$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.2.1.1.1.2

Rewrite the expression.

$5 (9 y^{2} - 15 y + 10) = 15 \cdot 2$

$x = \frac{5}{2} - \frac{3 y}{2}$

$5 (9 y^{2} - 15 y + 10) = 15 \cdot 2$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.2.1.1.2

Apply the distributive property.

$5 (9 y^{2}) + 5 (- 15 y) + 5 \cdot 10 = 15 \cdot 2$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.2.1.1.3

Simplify.

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Step 3.2.1.1.3.1

Multiply $9$ by $5$ .

$45 y^{2} + 5 (- 15 y) + 5 \cdot 10 = 15 \cdot 2$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.2.1.1.3.2

Multiply $- 15$ by $5$ .

$45 y^{2} - 75 y + 5 \cdot 10 = 15 \cdot 2$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.2.1.1.3.3

Multiply $5$ by $10$ .

$45 y^{2} - 75 y + 50 = 15 \cdot 2$

$x = \frac{5}{2} - \frac{3 y}{2}$

$45 y^{2} - 75 y + 50 = 15 \cdot 2$

$x = \frac{5}{2} - \frac{3 y}{2}$

$45 y^{2} - 75 y + 50 = 15 \cdot 2$

$x = \frac{5}{2} - \frac{3 y}{2}$

$45 y^{2} - 75 y + 50 = 15 \cdot 2$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $15$ by $2$ .

$45 y^{2} - 75 y + 50 = 30$

$x = \frac{5}{2} - \frac{3 y}{2}$

$45 y^{2} - 75 y + 50 = 30$

$x = \frac{5}{2} - \frac{3 y}{2}$

$45 y^{2} - 75 y + 50 = 30$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Subtract $30$ from both sides of the equation.

$45 y^{2} - 75 y + 50 - 30 = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.2

Subtract $30$ from $50$ .

$45 y^{2} - 75 y + 20 = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.3

Factor the left side of the equation.

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Step 3.3.3.1

Factor $5$ out of $45 y^{2} - 75 y + 20$ .

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Step 3.3.3.1.1

Factor $5$ out of $45 y^{2}$ .

$5 (9 y^{2}) - 75 y + 20 = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.3.1.2

Factor $5$ out of $- 75 y$ .

$5 (9 y^{2}) + 5 (- 15 y) + 20 = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.3.1.3

Factor $5$ out of $20$ .

$5 (9 y^{2}) + 5 (- 15 y) + 5 (4) = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.3.1.4

Factor $5$ out of $5 (9 y^{2}) + 5 (- 15 y)$ .

$5 (9 y^{2} - 15 y) + 5 (4) = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.3.1.5

Factor $5$ out of $5 (9 y^{2} - 15 y) + 5 (4)$ .

$5 (9 y^{2} - 15 y + 4) = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

$5 (9 y^{2} - 15 y + 4) = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.3.2

Factor.

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Step 3.3.3.2.1

Factor by grouping.

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Step 3.3.3.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 9 \cdot 4 = 36$ and whose sum is $b = - 15$ .

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Step 3.3.3.2.1.1.1

Factor $- 15$ out of $- 15 y$ .

$5 (9 y^{2} - 15 y + 4) = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.3.2.1.1.2

Rewrite $- 15$ as $- 3$ plus $- 12$

$5 (9 y^{2} + (- 3 - 12) y + 4) = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.3.2.1.1.3

Apply the distributive property.

$5 (9 y^{2} - 3 y - 12 y + 4) = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

$5 (9 y^{2} - 3 y - 12 y + 4) = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.3.2.1.2

Factor out the greatest common factor from each group.

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Step 3.3.3.2.1.2.1

Group the first two terms and the last two terms.

$5 ((9 y^{2} - 3 y) - 12 y + 4) = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.3.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$5 (3 y (3 y - 1) - 4 (3 y - 1)) = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

$5 (3 y (3 y - 1) - 4 (3 y - 1)) = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.3.2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 y - 1$ .

$5 ((3 y - 1) (3 y - 4)) = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

$5 ((3 y - 1) (3 y - 4)) = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.3.2.2

Remove unnecessary parentheses.

$5 (3 y - 1) (3 y - 4) = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

$5 (3 y - 1) (3 y - 4) = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

$5 (3 y - 1) (3 y - 4) = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.4

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

$3 y - 1 = 0$

$3 y - 4 = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.5

Set $3 y - 1$ equal to $0$ and solve for $y$ .

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Step 3.3.5.1

Set $3 y - 1$ equal to $0$ .

$3 y - 1 = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.5.2

Solve $3 y - 1 = 0$ for $y$ .

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Step 3.3.5.2.1

Add $1$ to both sides of the equation.

$3 y = 1$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.5.2.2

Divide each term in $3 y = 1$ by $3$ and simplify.

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Step 3.3.5.2.2.1

Divide each term in $3 y = 1$ by $3$ .

$\frac{3 y}{3} = \frac{1}{3}$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.5.2.2.2

Simplify the left side.

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Step 3.3.5.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.3.5.2.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{1}{3}$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.5.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{1}{3}$

$x = \frac{5}{2} - \frac{3 y}{2}$

$y = \frac{1}{3}$

$x = \frac{5}{2} - \frac{3 y}{2}$

$y = \frac{1}{3}$

$x = \frac{5}{2} - \frac{3 y}{2}$

$y = \frac{1}{3}$

$x = \frac{5}{2} - \frac{3 y}{2}$

$y = \frac{1}{3}$

$x = \frac{5}{2} - \frac{3 y}{2}$

$y = \frac{1}{3}$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.6

Set $3 y - 4$ equal to $0$ and solve for $y$ .

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Step 3.3.6.1

Set $3 y - 4$ equal to $0$ .

$3 y - 4 = 0$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.6.2

Solve $3 y - 4 = 0$ for $y$ .

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Step 3.3.6.2.1

Add $4$ to both sides of the equation.

$3 y = 4$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.6.2.2

Divide each term in $3 y = 4$ by $3$ and simplify.

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Step 3.3.6.2.2.1

Divide each term in $3 y = 4$ by $3$ .

$\frac{3 y}{3} = \frac{4}{3}$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.6.2.2.2

Simplify the left side.

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Step 3.3.6.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.3.6.2.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{4}{3}$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.6.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{4}{3}$

$x = \frac{5}{2} - \frac{3 y}{2}$

$y = \frac{4}{3}$

$x = \frac{5}{2} - \frac{3 y}{2}$

$y = \frac{4}{3}$

$x = \frac{5}{2} - \frac{3 y}{2}$

$y = \frac{4}{3}$

$x = \frac{5}{2} - \frac{3 y}{2}$

$y = \frac{4}{3}$

$x = \frac{5}{2} - \frac{3 y}{2}$

$y = \frac{4}{3}$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 3.3.7

The final solution is all the values that make $5 (3 y - 1) (3 y - 4) = 0$ true.

$y = \frac{1}{3}, \frac{4}{3}$

$x = \frac{5}{2} - \frac{3 y}{2}$

$y = \frac{1}{3}, \frac{4}{3}$

$x = \frac{5}{2} - \frac{3 y}{2}$

$y = \frac{1}{3}, \frac{4}{3}$

$x = \frac{5}{2} - \frac{3 y}{2}$

Step 4

Replace all occurrences of $y$ with $\frac{1}{3}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{5}{2} - \frac{3 y}{2}$ with $\frac{1}{3}$ .

$x = \frac{5}{2} - \frac{3 (\frac{1}{3})}{2}$

$y = \frac{1}{3}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{5}{2} - \frac{3 (\frac{1}{3})}{2}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{5 - 3 (\frac{1}{3})}{2}$

$y = \frac{1}{3}$

Step 4.2.1.2

Cancel the common factor of $3$ .

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Step 4.2.1.2.1

Factor $3$ out of $- 3$ .

$x = \frac{5 + 3 (- 1) (\frac{1}{3})}{2}$

$y = \frac{1}{3}$

Step 4.2.1.2.2

Cancel the common factor.

$x = \frac{5 + 3 \cdot (- 1 (\frac{1}{3}))}{2}$

$y = \frac{1}{3}$

Step 4.2.1.2.3

Rewrite the expression.

$x = \frac{5 - 1}{2}$

$y = \frac{1}{3}$

$x = \frac{5 - 1}{2}$

$y = \frac{1}{3}$

Step 4.2.1.3

Simplify the expression.

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Step 4.2.1.3.1

Subtract $1$ from $5$ .

$x = \frac{4}{2}$

$y = \frac{1}{3}$

Step 4.2.1.3.2

Divide $4$ by $2$ .

$x = 2$

$y = \frac{1}{3}$

$x = 2$

$y = \frac{1}{3}$

$x = 2$

$y = \frac{1}{3}$

$x = 2$

$y = \frac{1}{3}$

$x = 2$

$y = \frac{1}{3}$

Step 5

Replace all occurrences of $y$ with $\frac{4}{3}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = \frac{5}{2} - \frac{3 y}{2}$ with $\frac{4}{3}$ .

$x = \frac{5}{2} - \frac{3 (\frac{4}{3})}{2}$

$y = \frac{4}{3}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{5}{2} - \frac{3 (\frac{4}{3})}{2}$ .

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Step 5.2.1.1

Combine the numerators over the common denominator.

$x = \frac{5 - 3 (\frac{4}{3})}{2}$

$y = \frac{4}{3}$

Step 5.2.1.2

Simplify each term.

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Step 5.2.1.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.2.1.1

Factor $3$ out of $- 3$ .

$x = \frac{5 + 3 (- 1) (\frac{4}{3})}{2}$

$y = \frac{4}{3}$

Step 5.2.1.2.1.2

Cancel the common factor.

$x = \frac{5 + 3 \cdot (- 1 (\frac{4}{3}))}{2}$

$y = \frac{4}{3}$

Step 5.2.1.2.1.3

Rewrite the expression.

$x = \frac{5 - 1 \cdot 4}{2}$

$y = \frac{4}{3}$

$x = \frac{5 - 1 \cdot 4}{2}$

$y = \frac{4}{3}$

Step 5.2.1.2.2

Multiply $- 1$ by $4$ .

$x = \frac{5 - 4}{2}$

$y = \frac{4}{3}$

$x = \frac{5 - 4}{2}$

$y = \frac{4}{3}$

Step 5.2.1.3

Subtract $4$ from $5$ .

$x = \frac{1}{2}$

$y = \frac{4}{3}$

$x = \frac{1}{2}$

$y = \frac{4}{3}$

$x = \frac{1}{2}$

$y = \frac{4}{3}$

$x = \frac{1}{2}$

$y = \frac{4}{3}$

Step 6

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

$(2, \frac{1}{3})$

$(\frac{1}{2}, \frac{4}{3})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(2, \frac{1}{3}), (\frac{1}{2}, \frac{4}{3})$

Equation Form:

$x = 2, y = \frac{1}{3}$

$x = \frac{1}{2}, y = \frac{4}{3}$

Step 8