Algebra Examples

$x^{2} + y^{2} = 17$ $y = - \frac{1}{2} x$

Step 1

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

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

$x^{2} + y^{2} = 17$

Step 2

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

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Step 2.1

Replace all occurrences of $y$ in $x^{2} + y^{2} = 17$ with $- \frac{x}{2}$ .

$x^{2} + {(- \frac{x}{2})}^{2} = 17$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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 $- \frac{x}{2}$ .

$x^{2} + {(- 1)}^{2} {(\frac{x}{2})}^{2} = 17$

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

Step 2.2.1.1.1.2

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

$x^{2} + {(- 1)}^{2} (\frac{x^{2}}{2^{2}}) = 17$

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

$x^{2} + {(- 1)}^{2} (\frac{x^{2}}{2^{2}}) = 17$

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

Step 2.2.1.1.2

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

$x^{2} + 1 (\frac{x^{2}}{2^{2}}) = 17$

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

Step 2.2.1.1.3

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

$x^{2} + \frac{x^{2}}{2^{2}} = 17$

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

Step 2.2.1.1.4

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

$x^{2} + \frac{x^{2}}{4} = 17$

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

$x^{2} + \frac{x^{2}}{4} = 17$

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

Step 2.2.1.2

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

$x^{2} \cdot \frac{4}{4} + \frac{x^{2}}{4} = 17$

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

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

Combine $x^{2}$ and $\frac{4}{4}$ .

$\frac{x^{2} \cdot 4}{4} + \frac{x^{2}}{4} = 17$

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

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 4 + x^{2}}{4} = 17$

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

$\frac{x^{2} \cdot 4 + x^{2}}{4} = 17$

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

Step 2.2.1.4

Simplify the numerator.

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Step 2.2.1.4.1

Move $4$ to the left of $x^{2}$ .

$\frac{4 \cdot x^{2} + x^{2}}{4} = 17$

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

Step 2.2.1.4.2

Add $4 x^{2}$ and $x^{2}$ .

$\frac{5 x^{2}}{4} = 17$

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

$\frac{5 x^{2}}{4} = 17$

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

$\frac{5 x^{2}}{4} = 17$

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

$\frac{5 x^{2}}{4} = 17$

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

$\frac{5 x^{2}}{4} = 17$

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

Step 3

Solve for $x$ in $\frac{5 x^{2}}{4} = 17$ .

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Step 3.1

Multiply both sides of the equation by $\frac{4}{5}$ .

$\frac{4}{5} \cdot \frac{5 x^{2}}{4} = \frac{4}{5} \cdot 17$

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

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 $\frac{4}{5} \cdot \frac{5 x^{2}}{4}$ .

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Step 3.2.1.1.1

Combine.

$\frac{4 (5 x^{2})}{5 \cdot 4} = \frac{4}{5} \cdot 17$

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

Step 3.2.1.1.2

Cancel the common factor of $4$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$\frac{4 (5 x^{2})}{5 \cdot 4} = \frac{4}{5} \cdot 17$

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

Step 3.2.1.1.2.2

Rewrite the expression.

$\frac{5 x^{2}}{5} = \frac{4}{5} \cdot 17$

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

$\frac{5 x^{2}}{5} = \frac{4}{5} \cdot 17$

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

Step 3.2.1.1.3

Cancel the common factor of $5$ .

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Step 3.2.1.1.3.1

Cancel the common factor.

$\frac{5 x^{2}}{5} = \frac{4}{5} \cdot 17$

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

Step 3.2.1.1.3.2

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

$x^{2} = \frac{4}{5} \cdot 17$

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

$x^{2} = \frac{4}{5} \cdot 17$

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

$x^{2} = \frac{4}{5} \cdot 17$

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

$x^{2} = \frac{4}{5} \cdot 17$

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $\frac{4}{5} \cdot 17$ .

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Step 3.2.2.1.1

Combine $\frac{4}{5}$ and $17$ .

$x^{2} = \frac{4 \cdot 17}{5}$

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

Step 3.2.2.1.2

Multiply $4$ by $17$ .

$x^{2} = \frac{68}{5}$

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

$x^{2} = \frac{68}{5}$

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

$x^{2} = \frac{68}{5}$

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

$x^{2} = \frac{68}{5}$

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

Step 3.3

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

$x = \pm \sqrt{\frac{68}{5}}$

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

Step 3.4

Simplify $\pm \sqrt{\frac{68}{5}}$ .

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Step 3.4.1

Rewrite $\sqrt{\frac{68}{5}}$ as $\frac{\sqrt{68}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{68}}{\sqrt{5}}$

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

Step 3.4.2

Simplify the numerator.

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Step 3.4.2.1

Rewrite $68$ as $2^{2} \cdot 17$ .

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Step 3.4.2.1.1

Factor $4$ out of $68$ .

$x = \pm \frac{\sqrt{4 (17)}}{\sqrt{5}}$

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

Step 3.4.2.1.2

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

$x = \pm \frac{\sqrt{2^{2} \cdot 17}}{\sqrt{5}}$

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

$x = \pm \frac{\sqrt{2^{2} \cdot 17}}{\sqrt{5}}$

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

Step 3.4.2.2

Pull terms out from under the radical.

$x = \pm \frac{2 \sqrt{17}}{\sqrt{5}}$

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

$x = \pm \frac{2 \sqrt{17}}{\sqrt{5}}$

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

Step 3.4.3

Multiply $\frac{2 \sqrt{17}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{2 \sqrt{17}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

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

Step 3.4.4

Combine and simplify the denominator.

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Step 3.4.4.1

Multiply $\frac{2 \sqrt{17}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{2 \sqrt{17} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

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

Step 3.4.4.2

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{2 \sqrt{17} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

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

Step 3.4.4.3

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{2 \sqrt{17} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

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

Step 3.4.4.4

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

$x = \pm \frac{2 \sqrt{17} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

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

Step 3.4.4.5

Add $1$ and $1$ .

$x = \pm \frac{2 \sqrt{17} \sqrt{5}}{{\sqrt{5}}^{2}}$

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

Step 3.4.4.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$x = \pm \frac{2 \sqrt{17} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

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

Step 3.4.4.6.2

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

$x = \pm \frac{2 \sqrt{17} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

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

Step 3.4.4.6.3

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

$x = \pm \frac{2 \sqrt{17} \sqrt{5}}{5^{\frac{2}{2}}}$

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

Step 3.4.4.6.4

Cancel the common factor of $2$ .

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Step 3.4.4.6.4.1

Cancel the common factor.

$x = \pm \frac{2 \sqrt{17} \sqrt{5}}{5^{\frac{2}{2}}}$

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

Step 3.4.4.6.4.2

Rewrite the expression.