Precalculus Examples

$x^{2} + 2 x y - y^{2} = 68$ , $x^{2} - y^{2} = - 28$

Step 1

Solve for $x$ in $x^{2} - y^{2} = - 28$ .

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Step 1.1

Add $y^{2}$ to both sides of the equation.

$x^{2} = - 28 + y^{2}$

$x^{2} + 2 x y - y^{2} = 68$

Step 1.2

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

$x = \pm \sqrt{- 28 + y^{2}}$

$x^{2} + 2 x y - y^{2} = 68$

Step 1.3

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

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Step 1.3.1

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

$x = \sqrt{- 28 + y^{2}}$

$x^{2} + 2 x y - y^{2} = 68$

Step 1.3.2

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

$x = - \sqrt{- 28 + y^{2}}$

$x^{2} + 2 x y - y^{2} = 68$

Step 1.3.3

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

$x = \sqrt{- 28 + y^{2}}$

$x = - \sqrt{- 28 + y^{2}}$

$x^{2} + 2 x y - y^{2} = 68$

$x = \sqrt{- 28 + y^{2}}$

$x = - \sqrt{- 28 + y^{2}}$

$x^{2} + 2 x y - y^{2} = 68$

$x = \sqrt{- 28 + y^{2}}$

$x = - \sqrt{- 28 + y^{2}}$

$x^{2} + 2 x y - y^{2} = 68$

Step 2

Solve the system $x = \sqrt{- 28 + y^{2}}, x^{2} + 2 x y - y^{2} = 68$ .

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Step 2.1

Replace all occurrences of $x$ with $\sqrt{- 28 + y^{2}}$ in each equation.

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Step 2.1.1

Replace all occurrences of $x$ in $x^{2} + 2 x y - y^{2} = 68$ with $\sqrt{- 28 + y^{2}}$ .

${(\sqrt{- 28 + y^{2}})}^{2} + 2 (\sqrt{- 28 + y^{2}}) \cdot y - y^{2} = 68$

$x = \sqrt{- 28 + y^{2}}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify ${(\sqrt{- 28 + y^{2}})}^{2} + 2 (\sqrt{- 28 + y^{2}}) \cdot y - y^{2}$ .

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Step 2.1.2.1.1

Rewrite ${\sqrt{- 28 + y^{2}}}^{2}$ as $- 28 + y^{2}$ .

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Step 2.1.2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- 28 + y^{2}}$ as ${(- 28 + y^{2})}^{\frac{1}{2}}$ .

${({(- 28 + y^{2})}^{\frac{1}{2}})}^{2} + 2 (\sqrt{- 28 + y^{2}}) \cdot y - y^{2} = 68$

$x = \sqrt{- 28 + y^{2}}$

Step 2.1.2.1.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(- 28 + y^{2})}^{\frac{1}{2} \cdot 2} + 2 (\sqrt{- 28 + y^{2}}) \cdot y - y^{2} = 68$

$x = \sqrt{- 28 + y^{2}}$

Step 2.1.2.1.1.3

Combine $\frac{1}{2}$ and $2$ .

${(- 28 + y^{2})}^{\frac{2}{2}} + 2 (\sqrt{- 28 + y^{2}}) \cdot y - y^{2} = 68$

$x = \sqrt{- 28 + y^{2}}$

Step 2.1.2.1.1.4

Cancel the common factor of $2$ .

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Step 2.1.2.1.1.4.1

Cancel the common factor.

${(- 28 + y^{2})}^{\frac{2}{2}} + 2 (\sqrt{- 28 + y^{2}}) \cdot y - y^{2} = 68$

$x = \sqrt{- 28 + y^{2}}$

Step 2.1.2.1.1.4.2

Rewrite the expression.

$(- 28 + y^{2}) + 2 (\sqrt{- 28 + y^{2}}) \cdot y - y^{2} = 68$

$x = \sqrt{- 28 + y^{2}}$

$(- 28 + y^{2}) + 2 (\sqrt{- 28 + y^{2}}) \cdot y - y^{2} = 68$

$x = \sqrt{- 28 + y^{2}}$

Step 2.1.2.1.1.5

Simplify.

$- 28 + y^{2} + 2 (\sqrt{- 28 + y^{2}}) \cdot y - y^{2} = 68$

$x = \sqrt{- 28 + y^{2}}$

$- 28 + y^{2} + 2 \sqrt{- 28 + y^{2}} y - y^{2} = 68$

$x = \sqrt{- 28 + y^{2}}$

Step 2.1.2.1.2

Combine the opposite terms in $- 28 + y^{2} + 2 \sqrt{- 28 + y^{2}} y - y^{2}$ .

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Step 2.1.2.1.2.1

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

$- 28 + 2 \sqrt{- 28 + y^{2}} y + 0 = 68$

$x = \sqrt{- 28 + y^{2}}$

Step 2.1.2.1.2.2

Add $- 28 + 2 \sqrt{- 28 + y^{2}} y$ and $0$ .

$- 28 + 2 \sqrt{- 28 + y^{2}} y = 68$

$x = \sqrt{- 28 + y^{2}}$

$- 28 + 2 \sqrt{- 28 + y^{2}} y = 68$

$x = \sqrt{- 28 + y^{2}}$

$- 28 + 2 \sqrt{- 28 + y^{2}} y = 68$

$x = \sqrt{- 28 + y^{2}}$

$- 28 + 2 \sqrt{- 28 + y^{2}} y = 68$

$x = \sqrt{- 28 + y^{2}}$

$- 28 + 2 \sqrt{- 28 + y^{2}} y = 68$

$x = \sqrt{- 28 + y^{2}}$

Step 2.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$y = 8$

$x = \sqrt{- 28 + y^{2}}$

Step 2.3

Replace all occurrences of $y$ with $8$ in each equation.

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Step 2.3.1

Replace all occurrences of $y$ in $x = \sqrt{- 28 + y^{2}}$ with $8$ .

$x = \sqrt{- 28 + {(8)}^{2}}$

$y = 8$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\sqrt{- 28 + {(8)}^{2}}$ .

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Step 2.3.2.1.1

Raise $8$ to the power of $2$ .

$x = \sqrt{- 28 + 64}$

$y = 8$

Step 2.3.2.1.2

Add $- 28$ and $64$ .

$x = \sqrt{36}$

$y = 8$

Step 2.3.2.1.3

Rewrite $36$ as $6^{2}$ .

$x = \sqrt{6^{2}}$

$y = 8$

Step 2.3.2.1.4

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

$x = 6$

$y = 8$

$x = 6$

$y = 8$

$x = 6$

$y = 8$

$x = 6$

$y = 8$

$x = 6$

$y = 8$

Step 3

Solve the system $x = - \sqrt{- 28 + y^{2}}, x^{2} + 2 x y - y^{2} = 68$ .

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Step 3.1

Replace all occurrences of $x$ with $- \sqrt{- 28 + y^{2}}$ in each equation.

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Step 3.1.1

Replace all occurrences of $x$ in $x^{2} + 2 x y - y^{2} = 68$ with $- \sqrt{- 28 + y^{2}}$ .

${(- \sqrt{- 28 + y^{2}})}^{2} + 2 (- \sqrt{- 28 + y^{2}}) \cdot y - y^{2} = 68$

$x = - \sqrt{- 28 + y^{2}}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Simplify ${(- \sqrt{- 28 + y^{2}})}^{2} + 2 (- \sqrt{- 28 + y^{2}}) \cdot y - y^{2}$ .

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Step 3.1.2.1.1

Simplify each term.

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Step 3.1.2.1.1.1

Apply the product rule to $- \sqrt{- 28 + y^{2}}$ .

${(- 1)}^{2} {\sqrt{- 28 + y^{2}}}^{2} + 2 (- \sqrt{- 28 + y^{2}}) \cdot y - y^{2} = 68$

$x = - \sqrt{- 28 + y^{2}}$

Step 3.1.2.1.1.2

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

$1 {\sqrt{- 28 + y^{2}}}^{2} + 2 (- \sqrt{- 28 + y^{2}}) \cdot y - y^{2} = 68$

$x = - \sqrt{- 28 + y^{2}}$

Step 3.1.2.1.1.3

Multiply ${\sqrt{- 28 + y^{2}}}^{2}$ by $1$ .

${\sqrt{- 28 + y^{2}}}^{2} + 2 (- \sqrt{- 28 + y^{2}}) \cdot y - y^{2} = 68$

$x = - \sqrt{- 28 + y^{2}}$

Step 3.1.2.1.1.4

Rewrite ${\sqrt{- 28 + y^{2}}}^{2}$ as $- 28 + y^{2}$ .

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Step 3.1.2.1.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- 28 + y^{2}}$ as ${(- 28 + y^{2})}^{\frac{1}{2}}$ .

${({(- 28 + y^{2})}^{\frac{1}{2}})}^{2} + 2 (- \sqrt{- 28 + y^{2}}) \cdot y - y^{2} = 68$

$x = - \sqrt{- 28 + y^{2}}$

Step 3.1.2.1.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(- 28 + y^{2})}^{\frac{1}{2} \cdot 2} + 2 (- \sqrt{- 28 + y^{2}}) \cdot y - y^{2} = 68$

$x = - \sqrt{- 28 + y^{2}}$

Step 3.1.2.1.1.4.3

Combine $\frac{1}{2}$ and $2$ .

${(- 28 + y^{2})}^{\frac{2}{2}} + 2 (- \sqrt{- 28 + y^{2}}) \cdot y - y^{2} = 68$

$x = - \sqrt{- 28 + y^{2}}$

Step 3.1.2.1.1.4.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.1.4.4.1

Cancel the common factor.

${(- 28 + y^{2})}^{\frac{2}{2}} + 2 (- \sqrt{- 28 + y^{2}}) \cdot y - y^{2} = 68$

$x = - \sqrt{- 28 + y^{2}}$

Step 3.1.2.1.1.4.4.2

Rewrite the expression.

$(- 28 + y^{2}) + 2 (- \sqrt{- 28 + y^{2}}) \cdot y - y^{2} = 68$

$x = - \sqrt{- 28 + y^{2}}$

$(- 28 + y^{2}) + 2 (- \sqrt{- 28 + y^{2}}) \cdot y - y^{2} = 68$

$x = - \sqrt{- 28 + y^{2}}$

Step 3.1.2.1.1.4.5

Simplify.

$- 28 + y^{2} + 2 (- \sqrt{- 28 + y^{2}}) \cdot y - y^{2} = 68$

$x = - \sqrt{- 28 + y^{2}}$

$- 28 + y^{2} + 2 (- \sqrt{- 28 + y^{2}}) \cdot y - y^{2} = 68$

$x = - \sqrt{- 28 + y^{2}}$

Step 3.1.2.1.1.5

Multiply $- 1$ by $2$ .

$- 28 + y^{2} - 2 \sqrt{- 28 + y^{2}} y - y^{2} = 68$

$x = - \sqrt{- 28 + y^{2}}$

$- 28 + y^{2} - 2 \sqrt{- 28 + y^{2}} y - y^{2} = 68$

$x = - \sqrt{- 28 + y^{2}}$

Step 3.1.2.1.2

Combine the opposite terms in $- 28 + y^{2} - 2 \sqrt{- 28 + y^{2}} y - y^{2}$ .

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Step 3.1.2.1.2.1

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

$- 28 - 2 \sqrt{- 28 + y^{2}} y + 0 = 68$

$x = - \sqrt{- 28 + y^{2}}$

Step 3.1.2.1.2.2

Add $- 28 - 2 \sqrt{- 28 + y^{2}} y$ and $0$ .