Algebra Examples

$3 y = 12 x$ $x^{2} + y^{2} = 81$

Step 1

Divide each term in $3 y = 12 x$ by $3$ and simplify.

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Step 1.1

Divide each term in $3 y = 12 x$ by $3$ .

$\frac{3 y}{3} = \frac{12 x}{3}$

$x^{2} + y^{2} = 81$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $3$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{12 x}{3}$

$x^{2} + y^{2} = 81$

Step 1.2.1.2

Divide $y$ by $1$ .

$y = \frac{12 x}{3}$

$x^{2} + y^{2} = 81$

$y = \frac{12 x}{3}$

$x^{2} + y^{2} = 81$

$y = \frac{12 x}{3}$

$x^{2} + y^{2} = 81$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $12$ and $3$ .

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Step 1.3.1.1

Factor $3$ out of $12 x$ .

$y = \frac{3 (4 x)}{3}$

$x^{2} + y^{2} = 81$

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $3$ out of $3$ .

$y = \frac{3 (4 x)}{3 (1)}$

$x^{2} + y^{2} = 81$

Step 1.3.1.2.2

Cancel the common factor.

$y = \frac{3 (4 x)}{3 \cdot 1}$

$x^{2} + y^{2} = 81$

Step 1.3.1.2.3

Rewrite the expression.

$y = \frac{4 x}{1}$

$x^{2} + y^{2} = 81$

Step 1.3.1.2.4

Divide $4 x$ by $1$ .

$y = 4 x$

$x^{2} + y^{2} = 81$

$y = 4 x$

$x^{2} + y^{2} = 81$

$y = 4 x$

$x^{2} + y^{2} = 81$

$y = 4 x$

$x^{2} + y^{2} = 81$

$y = 4 x$

$x^{2} + y^{2} = 81$

Step 2

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

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Step 2.1

Replace all occurrences of $y$ in $x^{2} + y^{2} = 81$ with $4 x$ .

$x^{2} + {(4 x)}^{2} = 81$

$y = 4 x$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $x^{2} + {(4 x)}^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the product rule to $4 x$ .

$x^{2} + 4^{2} x^{2} = 81$

$y = 4 x$

Step 2.2.1.1.2

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

$x^{2} + 16 x^{2} = 81$

$y = 4 x$

$x^{2} + 16 x^{2} = 81$

$y = 4 x$

Step 2.2.1.2

Add $x^{2}$ and $16 x^{2}$ .

$17 x^{2} = 81$

$y = 4 x$

$17 x^{2} = 81$

$y = 4 x$

$17 x^{2} = 81$

$y = 4 x$

$17 x^{2} = 81$

$y = 4 x$

Step 3

Solve for $x$ in $17 x^{2} = 81$ .

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Step 3.1

Divide each term in $17 x^{2} = 81$ by $17$ and simplify.

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Step 3.1.1

Divide each term in $17 x^{2} = 81$ by $17$ .

$\frac{17 x^{2}}{17} = \frac{81}{17}$

$y = 4 x$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $17$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{17 x^{2}}{17} = \frac{81}{17}$

$y = 4 x$

Step 3.1.2.1.2

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

$x^{2} = \frac{81}{17}$

$y = 4 x$

$x^{2} = \frac{81}{17}$

$y = 4 x$

$x^{2} = \frac{81}{17}$

$y = 4 x$

$x^{2} = \frac{81}{17}$

$y = 4 x$

Step 3.2

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

$x = \pm \sqrt{\frac{81}{17}}$

$y = 4 x$

Step 3.3

Simplify $\pm \sqrt{\frac{81}{17}}$ .

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Step 3.3.1

Rewrite $\sqrt{\frac{81}{17}}$ as $\frac{\sqrt{81}}{\sqrt{17}}$ .

$x = \pm \frac{\sqrt{81}}{\sqrt{17}}$

$y = 4 x$

Step 3.3.2

Simplify the numerator.

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Step 3.3.2.1

Rewrite $81$ as $9^{2}$ .

$x = \pm \frac{\sqrt{9^{2}}}{\sqrt{17}}$

$y = 4 x$

Step 3.3.2.2

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

$x = \pm \frac{9}{\sqrt{17}}$

$y = 4 x$

$x = \pm \frac{9}{\sqrt{17}}$

$y = 4 x$

Step 3.3.3

Multiply $\frac{9}{\sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$x = \pm \frac{9}{\sqrt{17}} \cdot \frac{\sqrt{17}}{\sqrt{17}}$

$y = 4 x$

Step 3.3.4

Combine and simplify the denominator.

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Step 3.3.4.1

Multiply $\frac{9}{\sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$x = \pm \frac{9 \sqrt{17}}{\sqrt{17} \sqrt{17}}$

$y = 4 x$

Step 3.3.4.2

Raise $\sqrt{17}$ to the power of $1$ .

$x = \pm \frac{9 \sqrt{17}}{\sqrt{17} \sqrt{17}}$

$y = 4 x$

Step 3.3.4.3

Raise $\sqrt{17}$ to the power of $1$ .

$x = \pm \frac{9 \sqrt{17}}{\sqrt{17} \sqrt{17}}$

$y = 4 x$

Step 3.3.4.4

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

$x = \pm \frac{9 \sqrt{17}}{{\sqrt{17}}^{1 + 1}}$

$y = 4 x$

Step 3.3.4.5

Add $1$ and $1$ .

$x = \pm \frac{9 \sqrt{17}}{{\sqrt{17}}^{2}}$

$y = 4 x$

Step 3.3.4.6

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

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Step 3.3.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$x = \pm \frac{9 \sqrt{17}}{{(17^{\frac{1}{2}})}^{2}}$

$y = 4 x$

Step 3.3.4.6.2

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

$x = \pm \frac{9 \sqrt{17}}{17^{\frac{1}{2} \cdot 2}}$

$y = 4 x$

Step 3.3.4.6.3

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

$x = \pm \frac{9 \sqrt{17}}{17^{\frac{2}{2}}}$

$y = 4 x$

Step 3.3.4.6.4

Cancel the common factor of $2$ .

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Step 3.3.4.6.4.1

Cancel the common factor.

$x = \pm \frac{9 \sqrt{17}}{17^{\frac{2}{2}}}$

$y = 4 x$

Step 3.3.4.6.4.2

Rewrite the expression.

$x = \pm \frac{9 \sqrt{17}}{17}$

$y = 4 x$

$x = \pm \frac{9 \sqrt{17}}{17}$

$y = 4 x$

Step 3.3.4.6.5

Evaluate the exponent.

$x = \pm \frac{9 \sqrt{17}}{17}$

$y = 4 x$

$x = \pm \frac{9 \sqrt{17}}{17}$

$y = 4 x$

$x = \pm \frac{9 \sqrt{17}}{17}$

$y = 4 x$

$x = \pm \frac{9 \sqrt{17}}{17}$

$y = 4 x$

Step 3.4

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

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Step 3.4.1

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

$x = \frac{9 \sqrt{17}}{17}$

$y = 4 x$

Step 3.4.2

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

$x = - \frac{9 \sqrt{17}}{17}$

$y = 4 x$

Step 3.4.3

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

$x = \frac{9 \sqrt{17}}{17}, - \frac{9 \sqrt{17}}{17}$

$y = 4 x$

$x = \frac{9 \sqrt{17}}{17}, - \frac{9 \sqrt{17}}{17}$

$y = 4 x$

$x = \frac{9 \sqrt{17}}{17}, - \frac{9 \sqrt{17}}{17}$

$y = 4 x$

Step 4

Replace all occurrences of $x$ with $\frac{9 \sqrt{17}}{17}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = 4 x$ with $\frac{9 \sqrt{17}}{17}$ .

$y = 4 (\frac{9 \sqrt{17}}{17})$

$x = \frac{9 \sqrt{17}}{17}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $4 (\frac{9 \sqrt{17}}{17})$ .

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Step 4.2.1.1

Combine $4$ and $\frac{9 \sqrt{17}}{17}$ .

$y = \frac{4 (9 \sqrt{17})}{17}$

$x = \frac{9 \sqrt{17}}{17}$

Step 4.2.1.2

Multiply $9$ by $4$ .

$y = \frac{36 \sqrt{17}}{17}$

$x = \frac{9 \sqrt{17}}{17}$

$y = \frac{36 \sqrt{17}}{17}$

$x = \frac{9 \sqrt{17}}{17}$

$y = \frac{36 \sqrt{17}}{17}$

$x = \frac{9 \sqrt{17}}{17}$

$y = \frac{36 \sqrt{17}}{17}$

$x = \frac{9 \sqrt{17}}{17}$

Step 5

Replace all occurrences of $x$ with $- \frac{9 \sqrt{17}}{17}$ in each equation.

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Step 5.1

Replace all occurrences of $x$ in $y = 4 x$ with $- \frac{9 \sqrt{17}}{17}$ .

$y = 4 (- \frac{9 \sqrt{17}}{17})$

$x = - \frac{9 \sqrt{17}}{17}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $4 (- \frac{9 \sqrt{17}}{17})$ .

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Step 5.2.1.1

Multiply $4 (- \frac{9 \sqrt{17}}{17})$ .

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Step 5.2.1.1.1

Multiply $- 1$ by $4$ .

$y = - 4 \frac{9 \sqrt{17}}{17}$

$x = - \frac{9 \sqrt{17}}{17}$

Step 5.2.1.1.2

Combine $- 4$ and $\frac{9 \sqrt{17}}{17}$ .

$y = \frac{- 4 (9 \sqrt{17})}{17}$

$x = - \frac{9 \sqrt{17}}{17}$

Step 5.2.1.1.3

Multiply $9$ by $- 4$ .

$y = \frac{- 36 \sqrt{17}}{17}$

$x = - \frac{9 \sqrt{17}}{17}$

$y = \frac{- 36 \sqrt{17}}{17}$

$x = - \frac{9 \sqrt{17}}{17}$

Step 5.2.1.2

Move the negative in front of the fraction.

$y = - \frac{36 \sqrt{17}}{17}$

$x = - \frac{9 \sqrt{17}}{17}$

$y = - \frac{36 \sqrt{17}}{17}$

$x = - \frac{9 \sqrt{17}}{17}$

$y = - \frac{36 \sqrt{17}}{17}$

$x = - \frac{9 \sqrt{17}}{17}$

$y = - \frac{36 \sqrt{17}}{17}$

$x = - \frac{9 \sqrt{17}}{17}$

Step 6

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

$(\frac{9 \sqrt{17}}{17}, \frac{36 \sqrt{17}}{17})$

$(- \frac{9 \sqrt{17}}{17}, - \frac{36 \sqrt{17}}{17})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(\frac{9 \sqrt{17}}{17}, \frac{36 \sqrt{17}}{17}), (- \frac{9 \sqrt{17}}{17}, - \frac{36 \sqrt{17}}{17})$

Equation Form:

$x = \frac{9 \sqrt{17}}{17}, y = \frac{36 \sqrt{17}}{17}$

$x = - \frac{9 \sqrt{17}}{17}, y = - \frac{36 \sqrt{17}}{17}$

Step 8