Algebra Examples

$3 x^{2} - 2 y^{2} = - 24$ $2 y = - 3 x$

Step 1

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

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Step 1.1

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

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

$3 x^{2} - 2 y^{2} = - 24$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

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

$3 x^{2} - 2 y^{2} = - 24$

Step 1.2.1.2

Divide $y$ by $1$ .

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

$3 x^{2} - 2 y^{2} = - 24$

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

$3 x^{2} - 2 y^{2} = - 24$

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

$3 x^{2} - 2 y^{2} = - 24$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative in front of the fraction.

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

$3 x^{2} - 2 y^{2} = - 24$

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

$3 x^{2} - 2 y^{2} = - 24$

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

$3 x^{2} - 2 y^{2} = - 24$

Step 2

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

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Step 2.1

Replace all occurrences of $y$ in $3 x^{2} - 2 y^{2} = - 24$ with $- \frac{3 x}{2}$ .

$3 x^{2} - 2 {(- \frac{3 x}{2})}^{2} = - 24$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 x^{2} - 2 {(- \frac{3 x}{2})}^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

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

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Step 2.2.1.1.1.1

Apply the product rule to $- \frac{3 x}{2}$ .

$3 x^{2} - 2 ({(- 1)}^{2} {(\frac{3 x}{2})}^{2}) = - 24$

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

Step 2.2.1.1.1.2

Apply the product rule to $\frac{3 x}{2}$ .

$3 x^{2} - 2 ({(- 1)}^{2} (\frac{{(3 x)}^{2}}{2^{2}})) = - 24$

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

Step 2.2.1.1.1.3

Apply the product rule to $3 x$ .

$3 x^{2} - 2 ({(- 1)}^{2} (\frac{3^{2} x^{2}}{2^{2}})) = - 24$

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

$3 x^{2} - 2 ({(- 1)}^{2} (\frac{3^{2} x^{2}}{2^{2}})) = - 24$

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

Step 2.2.1.1.2

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

$3 x^{2} - 2 (1 (\frac{3^{2} x^{2}}{2^{2}})) = - 24$

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

Step 2.2.1.1.3

Multiply $\frac{3^{2} x^{2}}{2^{2}}$ by $1$ .

$3 x^{2} - 2 \frac{3^{2} x^{2}}{2^{2}} = - 24$

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

Step 2.2.1.1.4

Raise $3$ to the power of $2$ .

$3 x^{2} - 2 \frac{9 x^{2}}{2^{2}} = - 24$

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

Step 2.2.1.1.5

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

$3 x^{2} - 2 \frac{9 x^{2}}{4} = - 24$

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

Step 2.2.1.1.6

Cancel the common factor of $2$ .

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Step 2.2.1.1.6.1

Factor $2$ out of $- 2$ .

$3 x^{2} + 2 (- 1) (\frac{9 x^{2}}{4}) = - 24$

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

Step 2.2.1.1.6.2

Factor $2$ out of $4$ .

$3 x^{2} + 2 \cdot (- 1 \frac{9 x^{2}}{2 \cdot 2}) = - 24$

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

Step 2.2.1.1.6.3

Cancel the common factor.

$3 x^{2} + 2 \cdot (- 1 \frac{9 x^{2}}{2 \cdot 2}) = - 24$

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

Step 2.2.1.1.6.4

Rewrite the expression.

$3 x^{2} - 1 \frac{9 x^{2}}{2} = - 24$

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

$3 x^{2} - 1 \frac{9 x^{2}}{2} = - 24$

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

Step 2.2.1.1.7

Rewrite $- 1 \frac{9 x^{2}}{2}$ as $- \frac{9 x^{2}}{2}$ .

$3 x^{2} - \frac{9 x^{2}}{2} = - 24$

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

$3 x^{2} - \frac{9 x^{2}}{2} = - 24$

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

Step 2.2.1.2

To write $3 x^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 x^{2} \cdot \frac{2}{2} - \frac{9 x^{2}}{2} = - 24$

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

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

Combine $3 x^{2}$ and $\frac{2}{2}$ .

$\frac{3 x^{2} \cdot 2}{2} - \frac{9 x^{2}}{2} = - 24$

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

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$\frac{3 x^{2} \cdot 2 - 9 x^{2}}{2} = - 24$

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

$\frac{3 x^{2} \cdot 2 - 9 x^{2}}{2} = - 24$

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

Step 2.2.1.4

Simplify the numerator.

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Step 2.2.1.4.1

Factor $3 x^{2}$ out of $3 x^{2} \cdot 2 - 9 x^{2}$ .

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Step 2.2.1.4.1.1

Factor $3 x^{2}$ out of $3 x^{2} \cdot 2$ .

$\frac{3 x^{2} (2) - 9 x^{2}}{2} = - 24$

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

Step 2.2.1.4.1.2

Factor $3 x^{2}$ out of $- 9 x^{2}$ .

$\frac{3 x^{2} (2) + 3 x^{2} (- 3)}{2} = - 24$

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

Step 2.2.1.4.1.3

Factor $3 x^{2}$ out of $3 x^{2} (2) + 3 x^{2} (- 3)$ .

$\frac{3 x^{2} (2 - 3)}{2} = - 24$

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

$\frac{3 x^{2} (2 - 3)}{2} = - 24$

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

Step 2.2.1.4.2

Subtract $3$ from $2$ .

$\frac{3 x^{2} \cdot - 1}{2} = - 24$

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

Step 2.2.1.4.3

Multiply $- 1$ by $3$ .

$\frac{- 3 x^{2}}{2} = - 24$

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

$\frac{- 3 x^{2}}{2} = - 24$

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

Step 2.2.1.5

Move the negative in front of the fraction.

$- \frac{3 x^{2}}{2} = - 24$

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

$- \frac{3 x^{2}}{2} = - 24$

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

$- \frac{3 x^{2}}{2} = - 24$

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

$- \frac{3 x^{2}}{2} = - 24$

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

Step 3

Solve for $x$ in $- \frac{3 x^{2}}{2} = - 24$ .

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Step 3.1

Multiply both sides of the equation by $- \frac{2}{3}$ .

$- \frac{2}{3} \cdot (- \frac{3 x^{2}}{2}) = - \frac{2}{3} \cdot - 24$

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

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $- \frac{2}{3} (- \frac{3 x^{2}}{2})$ .

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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

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$\frac{- 2}{3} \cdot (- \frac{3 x^{2}}{2}) = - \frac{2}{3} \cdot - 24$

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

Step 3.2.1.1.1.2

Move the leading negative in $- \frac{3 x^{2}}{2}$ into the numerator.

$\frac{- 2}{3} \cdot \frac{- 3 x^{2}}{2} = - \frac{2}{3} \cdot - 24$

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

Step 3.2.1.1.1.3

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{3} \cdot \frac{- 3 x^{2}}{2} = - \frac{2}{3} \cdot - 24$

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

Step 3.2.1.1.1.4

Cancel the common factor.

$\frac{2 \cdot - 1}{3} \cdot \frac{- 3 x^{2}}{2} = - \frac{2}{3} \cdot - 24$

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

Step 3.2.1.1.1.5

Rewrite the expression.

$\frac{- 1}{3} \cdot (- 3 x^{2}) = - \frac{2}{3} \cdot - 24$

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

$\frac{- 1}{3} \cdot (- 3 x^{2}) = - \frac{2}{3} \cdot - 24$

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

Step 3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.1.2.1

Factor $3$ out of $- 3 x^{2}$ .

$\frac{- 1}{3} \cdot (3 (- x^{2})) = - \frac{2}{3} \cdot - 24$

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

Step 3.2.1.1.2.2

Cancel the common factor.

$\frac{- 1}{3} \cdot (3 (- x^{2})) = - \frac{2}{3} \cdot - 24$

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

Step 3.2.1.1.2.3

Rewrite the expression.

$x^{2} = - \frac{2}{3} \cdot - 24$

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

$x^{2} = - \frac{2}{3} \cdot - 24$

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

Step 3.2.1.1.3

Multiply.

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Step 3.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x^{2} = - \frac{2}{3} \cdot - 24$

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

Step 3.2.1.1.3.2

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

$x^{2} = - \frac{2}{3} \cdot - 24$

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

$x^{2} = - \frac{2}{3} \cdot - 24$

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

$x^{2} = - \frac{2}{3} \cdot - 24$

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

$x^{2} = - \frac{2}{3} \cdot - 24$

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $- \frac{2}{3} \cdot - 24$ .

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Step 3.2.2.1.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$x^{2} = \frac{- 2}{3} \cdot - 24$

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

Step 3.2.2.1.1.2

Factor $3$ out of $- 24$ .

$x^{2} = \frac{- 2}{3} \cdot (3 (- 8))$

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

Step 3.2.2.1.1.3

Cancel the common factor.

$x^{2} = \frac{- 2}{3} \cdot (3 \cdot - 8)$

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

Step 3.2.2.1.1.4

Rewrite the expression.

$x^{2} = - 2 \cdot - 8$

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

$x^{2} = - 2 \cdot - 8$

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

Step 3.2.2.1.2

Multiply $- 2$ by $- 8$ .

$x^{2} = 16$

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

$x^{2} = 16$

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

$x^{2} = 16$

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

$x^{2} = 16$

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

Step 3.3

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

$x = \pm \sqrt{16}$

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

Step 3.4

Simplify $\pm \sqrt{16}$ .

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Step 3.4.1

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

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

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

Step 3.4.2

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

$x = \pm 4$

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

$x = \pm 4$

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

Step 3.5

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

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Step 3.5.1

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

$x = 4$

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

Step 3.5.2

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

$x = - 4$

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

Step 3.5.3

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

$x = 4, - 4$

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

$x = 4, - 4$

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

$x = 4, - 4$

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

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 $y = - \frac{3 x}{2}$ with $4$ .

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

$x = 4$

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

Cancel the common factor of $4$ and $2$ .

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Step 4.2.1.1.1

Factor $2$ out of $3 (4)$ .

$y = - \frac{2 (3 \cdot (2))}{2}$

$x = 4$

Step 4.2.1.1.2

Cancel the common factors.

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Step 4.2.1.1.2.1

Factor $2$ out of $2$ .

$y = - \frac{2 (3 \cdot (2))}{2 (1)}$

$x = 4$

Step 4.2.1.1.2.2

Cancel the common factor.

$y = - \frac{2 (3 \cdot (2))}{2 \cdot 1}$

$x = 4$

Step 4.2.1.1.2.3

Rewrite the expression.

$y = - \frac{3 \cdot (2)}{1}$

$x = 4$

Step 4.2.1.1.2.4

Divide $3 \cdot (2)$ by $1$ .

$y = - (3 \cdot (2))$

$x = 4$

$y = - (3 \cdot (2))$

$x = 4$

$y = - (3 \cdot (2))$

$x = 4$

Step 4.2.1.2

Multiply $- (3 \cdot (2))$ .

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Step 4.2.1.2.1

Multiply $3$ by $2$ .

$y = - 1 \cdot 6$

$x = 4$

Step 4.2.1.2.2

Multiply $- 1$ by $6$ .

$y = - 6$

$x = 4$

$y = - 6$

$x = 4$

$y = - 6$

$x = 4$

$y = - 6$

$x = 4$

$y = - 6$

$x = 4$

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 $y = - \frac{3 x}{2}$ with $- 4$ .

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

$x = - 4$

Step 5.2

Simplify the right side.

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Step 5.2.1

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

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Step 5.2.1.1

Cancel the common factor of $- 4$ and $2$ .

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Step 5.2.1.1.1

Factor $2$ out of $3 (- 4)$ .

$y = - \frac{2 (3 \cdot (- 2))}{2}$

$x = - 4$

Step 5.2.1.1.2

Cancel the common factors.

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Step 5.2.1.1.2.1

Factor $2$ out of $2$ .

$y = - \frac{2 (3 \cdot (- 2))}{2 (1)}$

$x = - 4$

Step 5.2.1.1.2.2

Cancel the common factor.

$y = - \frac{2 (3 \cdot (- 2))}{2 \cdot 1}$

$x = - 4$

Step 5.2.1.1.2.3

Rewrite the expression.

$y = - \frac{3 \cdot (- 2)}{1}$

$x = - 4$

Step 5.2.1.1.2.4

Divide $3 \cdot (- 2)$ by $1$ .

$y = - (3 \cdot (- 2))$

$x = - 4$

$y = - (3 \cdot (- 2))$

$x = - 4$

$y = - (3 \cdot (- 2))$

$x = - 4$

Step 5.2.1.2

Multiply $- (3 \cdot (- 2))$ .

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Step 5.2.1.2.1

Multiply $3$ by $- 2$ .

$y = 6$

$x = - 4$

Step 5.2.1.2.2

Multiply $- 1$ by $- 6$ .

$y = 6$

$x = - 4$

$y = 6$

$x = - 4$

$y = 6$

$x = - 4$

$y = 6$

$x = - 4$

$y = 6$

$x = - 4$

Step 6

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

$(4, - 6)$

$(- 4, 6)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(4, - 6), (- 4, 6)$

Equation Form:

$x = 4, y = - 6$

$x = - 4, y = 6$

Step 8