Algebra Examples

$8 y = - 10 x$ $y^{2} = 2 x^{2} - 7$

Step 1

Divide each term in $8 y = - 10 x$ by $8$ and simplify.

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Step 1.1

Divide each term in $8 y = - 10 x$ by $8$ .

$\frac{8 y}{8} = \frac{- 10 x}{8}$

$y^{2} = 2 x^{2} - 7$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $8$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{8 y}{8} = \frac{- 10 x}{8}$

$y^{2} = 2 x^{2} - 7$

Step 1.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 10 x}{8}$

$y^{2} = 2 x^{2} - 7$

$y = \frac{- 10 x}{8}$

$y^{2} = 2 x^{2} - 7$

$y = \frac{- 10 x}{8}$

$y^{2} = 2 x^{2} - 7$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $- 10$ and $8$ .

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Step 1.3.1.1

Factor $2$ out of $- 10 x$ .

$y = \frac{2 (- 5 x)}{8}$

$y^{2} = 2 x^{2} - 7$

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $2$ out of $8$ .

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

$y^{2} = 2 x^{2} - 7$

Step 1.3.1.2.2

Cancel the common factor.

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

$y^{2} = 2 x^{2} - 7$

Step 1.3.1.2.3

Rewrite the expression.

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

$y^{2} = 2 x^{2} - 7$

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

$y^{2} = 2 x^{2} - 7$

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

$y^{2} = 2 x^{2} - 7$

Step 1.3.2

Move the negative in front of the fraction.

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

$y^{2} = 2 x^{2} - 7$

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

$y^{2} = 2 x^{2} - 7$

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

$y^{2} = 2 x^{2} - 7$

Step 2

Replace all occurrences of $y$ with $- \frac{5 x}{4}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $y^{2} = 2 x^{2} - 7$ with $- \frac{5 x}{4}$ .

${(- \frac{5 x}{4})}^{2} = 2 x^{2} - 7$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.2.1.1.1

Apply the product rule to $- \frac{5 x}{4}$ .

${(- 1)}^{2} {(\frac{5 x}{4})}^{2} = 2 x^{2} - 7$

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

Step 2.2.1.1.2

Apply the product rule to $\frac{5 x}{4}$ .

${(- 1)}^{2} (\frac{{(5 x)}^{2}}{4^{2}}) = 2 x^{2} - 7$

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

Step 2.2.1.1.3

Apply the product rule to $5 x$ .

${(- 1)}^{2} (\frac{5^{2} x^{2}}{4^{2}}) = 2 x^{2} - 7$

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

${(- 1)}^{2} (\frac{5^{2} x^{2}}{4^{2}}) = 2 x^{2} - 7$

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

Step 2.2.1.2

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

$1 (\frac{5^{2} x^{2}}{4^{2}}) = 2 x^{2} - 7$

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

Step 2.2.1.3

Multiply $\frac{5^{2} x^{2}}{4^{2}}$ by $1$ .

$\frac{5^{2} x^{2}}{4^{2}} = 2 x^{2} - 7$

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

Step 2.2.1.4

Raise $5$ to the power of $2$ .

$\frac{25 x^{2}}{4^{2}} = 2 x^{2} - 7$

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

Step 2.2.1.5

Raise $4$ to the power of $2$ .

$\frac{25 x^{2}}{16} = 2 x^{2} - 7$

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

$\frac{25 x^{2}}{16} = 2 x^{2} - 7$

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

$\frac{25 x^{2}}{16} = 2 x^{2} - 7$

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

$\frac{25 x^{2}}{16} = 2 x^{2} - 7$

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

Step 3

Solve for $x$ in $\frac{25 x^{2}}{16} = 2 x^{2} - 7$ .

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Step 3.1

Multiply both sides by $16$ .

$\frac{25 x^{2}}{16} \cdot 16 = (2 x^{2} - 7) \cdot 16$

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

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

Cancel the common factor of $16$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{25 x^{2}}{16} \cdot 16 = (2 x^{2} - 7) \cdot 16$

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

Step 3.2.1.1.2

Rewrite the expression.

$25 x^{2} = (2 x^{2} - 7) \cdot 16$

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

$25 x^{2} = (2 x^{2} - 7) \cdot 16$

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

$25 x^{2} = (2 x^{2} - 7) \cdot 16$

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $(2 x^{2} - 7) \cdot 16$ .

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Step 3.2.2.1.1

Apply the distributive property.

$25 x^{2} = 2 x^{2} \cdot 16 - 7 \cdot 16$

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

Step 3.2.2.1.2

Multiply.

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Step 3.2.2.1.2.1

Multiply $16$ by $2$ .

$25 x^{2} = 32 x^{2} - 7 \cdot 16$

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

Step 3.2.2.1.2.2

Multiply $- 7$ by $16$ .

$25 x^{2} = 32 x^{2} - 112$

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

$25 x^{2} = 32 x^{2} - 112$

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

$25 x^{2} = 32 x^{2} - 112$

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

$25 x^{2} = 32 x^{2} - 112$

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

$25 x^{2} = 32 x^{2} - 112$

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

Step 3.3

Solve for $x$ .

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Step 3.3.1

Move all terms containing $x$ to the left side of the equation.

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Step 3.3.1.1

Subtract $32 x^{2}$ from both sides of the equation.

$25 x^{2} - 32 x^{2} = - 112$

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

Step 3.3.1.2

Subtract $32 x^{2}$ from $25 x^{2}$ .

$- 7 x^{2} = - 112$

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

$- 7 x^{2} = - 112$

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

Step 3.3.2

Divide each term in $- 7 x^{2} = - 112$ by $- 7$ and simplify.

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Step 3.3.2.1

Divide each term in $- 7 x^{2} = - 112$ by $- 7$ .

$\frac{- 7 x^{2}}{- 7} = \frac{- 112}{- 7}$

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

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $- 7$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{- 7 x^{2}}{- 7} = \frac{- 112}{- 7}$

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

Step 3.3.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 112}{- 7}$

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

$x^{2} = \frac{- 112}{- 7}$

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

$x^{2} = \frac{- 112}{- 7}$

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

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Divide $- 112$ by $- 7$ .

$x^{2} = 16$

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

$x^{2} = 16$

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

$x^{2} = 16$

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

Step 3.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{16}$

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

Step 3.3.4

Simplify $\pm \sqrt{16}$ .

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Step 3.3.4.1

Rewrite $16$ as $4^{2}$ .

$x = \pm \sqrt{4^{2}}$

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

Step 3.3.4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 4$

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

$x = \pm 4$

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

Step 3.3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 4$

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

Step 3.3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 4$

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

Step 3.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 4, - 4$

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

$x = 4, - 4$

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

$x = 4, - 4$

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

$x = 4, - 4$

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

Step 4

Replace all occurrences of $x$ with $4$ in each equation.

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Step 4.1

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

$y = - \frac{5 (4)}{4}$

$x = 4$

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

Cancel the common factor of $4$ .

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Step 4.2.1.1.1

Cancel the common factor.

$y = - \frac{5 \cdot 4}{4}$

$x = 4$

Step 4.2.1.1.2

Divide $5$ by $1$ .

$y = - 1 \cdot 5$

$x = 4$

$y = - 1 \cdot 5$

$x = 4$

Step 4.2.1.2

Multiply $- 1$ by $5$ .

$y = - 5$

$x = 4$

$y = - 5$

$x = 4$

$y = - 5$

$x = 4$

$y = - 5$

$x = 4$

Step 5

Replace all occurrences of $x$ with $- 4$ in each equation.

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Step 5.1

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

$y = - \frac{5 (- 4)}{4}$

$x = - 4$

Step 5.2

Simplify the right side.

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Step 5.2.1

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

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Step 5.2.1.1

Cancel the common factor of $- 4$ and $4$ .

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Step 5.2.1.1.1

Factor $4$ out of $5 (- 4)$ .

$y = - \frac{4 (5 \cdot (- 1))}{4}$

$x = - 4$

Step 5.2.1.1.2

Cancel the common factors.

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Step 5.2.1.1.2.1

Factor $4$ out of $4$ .

$y = - \frac{4 (5 \cdot (- 1))}{4 (1)}$

$x = - 4$

Step 5.2.1.1.2.2

Cancel the common factor.

$y = - \frac{4 (5 \cdot (- 1))}{4 \cdot 1}$

$x = - 4$

Step 5.2.1.1.2.3

Rewrite the expression.

$y = - \frac{5 \cdot (- 1)}{1}$

$x = - 4$

Step 5.2.1.1.2.4

Divide $5 \cdot (- 1)$ by $1$ .

$y = - (5 \cdot (- 1))$

$x = - 4$

$y = - (5 \cdot (- 1))$

$x = - 4$

$y = - (5 \cdot (- 1))$

$x = - 4$

Step 5.2.1.2

Multiply $- (5 \cdot (- 1))$ .

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Step 5.2.1.2.1

Multiply $5$ by $- 1$ .

$y = 5$

$x = - 4$

Step 5.2.1.2.2

Multiply $- 1$ by $- 5$ .

$y = 5$

$x = - 4$

$y = 5$

$x = - 4$

$y = 5$

$x = - 4$

$y = 5$

$x = - 4$

$y = 5$

$x = - 4$

Step 6

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

$(4, - 5)$

$(- 4, 5)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(4, - 5), (- 4, 5)$

Equation Form:

$x = 4, y = - 5$

$x = - 4, y = 5$

Step 8