Precalculus Examples

$x^{2} + y^{2} = 100$ , $8 x - 6 y = 0$

Step 1

Solve for $x$ in $8 x - 6 y = 0$ .

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Step 1.1

Add $6 y$ to both sides of the equation.

$8 x = 6 y$

$x^{2} + y^{2} = 100$

Step 1.2

Divide each term in $8 x = 6 y$ by $8$ and simplify.

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Step 1.2.1

Divide each term in $8 x = 6 y$ by $8$ .

$\frac{8 x}{8} = \frac{6 y}{8}$

$x^{2} + y^{2} = 100$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $8$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{6 y}{8}$

$x^{2} + y^{2} = 100$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{6 y}{8}$

$x^{2} + y^{2} = 100$

$x = \frac{6 y}{8}$

$x^{2} + y^{2} = 100$

$x = \frac{6 y}{8}$

$x^{2} + y^{2} = 100$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Cancel the common factor of $6$ and $8$ .

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Step 1.2.3.1.1

Factor $2$ out of $6 y$ .

$x = \frac{2 (3 y)}{8}$

$x^{2} + y^{2} = 100$

Step 1.2.3.1.2

Cancel the common factors.

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Step 1.2.3.1.2.1

Factor $2$ out of $8$ .

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

$x^{2} + y^{2} = 100$

Step 1.2.3.1.2.2

Cancel the common factor.

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

$x^{2} + y^{2} = 100$

Step 1.2.3.1.2.3

Rewrite the expression.

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

$x^{2} + y^{2} = 100$

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

$x^{2} + y^{2} = 100$

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

$x^{2} + y^{2} = 100$

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

$x^{2} + y^{2} = 100$

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

$x^{2} + y^{2} = 100$

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

$x^{2} + y^{2} = 100$

Step 2

Replace all occurrences of $x$ with $\frac{3 y}{4}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x^{2} + y^{2} = 100$ with $\frac{3 y}{4}$ .

${(\frac{3 y}{4})}^{2} + y^{2} = 100$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(\frac{3 y}{4})}^{2} + y^{2}$ .

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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 $\frac{3 y}{4}$ .

$\frac{{(3 y)}^{2}}{4^{2}} + y^{2} = 100$

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

Step 2.2.1.1.1.2

Apply the product rule to $3 y$ .

$\frac{3^{2} y^{2}}{4^{2}} + y^{2} = 100$

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

$\frac{3^{2} y^{2}}{4^{2}} + y^{2} = 100$

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

Step 2.2.1.1.2

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

$\frac{9 y^{2}}{4^{2}} + y^{2} = 100$

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

Step 2.2.1.1.3

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

$\frac{9 y^{2}}{16} + y^{2} = 100$

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

$\frac{9 y^{2}}{16} + y^{2} = 100$

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

Step 2.2.1.2

To write $y^{2}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{9 y^{2}}{16} + y^{2} \cdot \frac{16}{16} = 100$

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

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

Combine $y^{2}$ and $\frac{16}{16}$ .

$\frac{9 y^{2}}{16} + \frac{y^{2} \cdot 16}{16} = 100$

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

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$\frac{9 y^{2} + y^{2} \cdot 16}{16} = 100$

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

$\frac{9 y^{2} + y^{2} \cdot 16}{16} = 100$

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

Step 2.2.1.4

Simplify the numerator.

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Step 2.2.1.4.1

Move $16$ to the left of $y^{2}$ .

$\frac{9 y^{2} + 16 \cdot y^{2}}{16} = 100$

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

Step 2.2.1.4.2

Add $9 y^{2}$ and $16 y^{2}$ .

$\frac{25 y^{2}}{16} = 100$

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

$\frac{25 y^{2}}{16} = 100$

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

$\frac{25 y^{2}}{16} = 100$

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

$\frac{25 y^{2}}{16} = 100$

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

$\frac{25 y^{2}}{16} = 100$

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

Step 3

Solve for $y$ in $\frac{25 y^{2}}{16} = 100$ .

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Step 3.1

Multiply both sides of the equation by $\frac{16}{25}$ .

$\frac{16}{25} \cdot \frac{25 y^{2}}{16} = \frac{16}{25} \cdot 100$

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

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 $\frac{16}{25} \cdot \frac{25 y^{2}}{16}$ .

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Step 3.2.1.1.1

Combine.

$\frac{16 (25 y^{2})}{25 \cdot 16} = \frac{16}{25} \cdot 100$

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

Step 3.2.1.1.2

Cancel the common factor of $16$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$\frac{16 (25 y^{2})}{25 \cdot 16} = \frac{16}{25} \cdot 100$

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

Step 3.2.1.1.2.2

Rewrite the expression.

$\frac{25 y^{2}}{25} = \frac{16}{25} \cdot 100$

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

$\frac{25 y^{2}}{25} = \frac{16}{25} \cdot 100$

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

Step 3.2.1.1.3

Cancel the common factor of $25$ .

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Step 3.2.1.1.3.1

Cancel the common factor.

$\frac{25 y^{2}}{25} = \frac{16}{25} \cdot 100$

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

Step 3.2.1.1.3.2

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

$y^{2} = \frac{16}{25} \cdot 100$

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

$y^{2} = \frac{16}{25} \cdot 100$

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

$y^{2} = \frac{16}{25} \cdot 100$

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

$y^{2} = \frac{16}{25} \cdot 100$

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\frac{16}{25} \cdot 100$ .

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Step 3.2.2.1.1

Cancel the common factor of $25$ .

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Step 3.2.2.1.1.1

Factor $25$ out of $100$ .

$y^{2} = \frac{16}{25} \cdot (25 (4))$

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

Step 3.2.2.1.1.2

Cancel the common factor.

$y^{2} = \frac{16}{25} \cdot (25 \cdot 4)$

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

Step 3.2.2.1.1.3

Rewrite the expression.

$y^{2} = 16 \cdot 4$

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

$y^{2} = 16 \cdot 4$

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

Step 3.2.2.1.2

Multiply $16$ by $4$ .

$y^{2} = 64$

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

$y^{2} = 64$

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

$y^{2} = 64$

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

$y^{2} = 64$

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

Step 3.3

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

$y = \pm \sqrt{64}$

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

Step 3.4

Simplify $\pm \sqrt{64}$ .

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Step 3.4.1

Rewrite $64$ as $8^{2}$ .

$y = \pm \sqrt{8^{2}}$

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

Step 3.4.2

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

$y = \pm 8$

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

$y = \pm 8$

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

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.

$y = 8$

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

Step 3.5.2

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

$y = - 8$

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

Step 3.5.3

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

$y = 8, - 8$

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

$y = 8, - 8$

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

$y = 8, - 8$

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

Step 4

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

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{3 y}{4}$ with $8$ .

$x = \frac{3 (8)}{4}$

$y = 8$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{3 (8)}{4}$ .

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Step 4.2.1.1

Cancel the common factor of $8$ and $4$ .

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Step 4.2.1.1.1

Factor $4$ out of $3 (8)$ .

$x = \frac{4 (3 \cdot (2))}{4}$

$y = 8$

Step 4.2.1.1.2

Cancel the common factors.

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Step 4.2.1.1.2.1

Factor $4$ out of $4$ .

$x = \frac{4 (3 \cdot (2))}{4 (1)}$

$y = 8$

Step 4.2.1.1.2.2

Cancel the common factor.

$x = \frac{4 (3 \cdot (2))}{4 \cdot 1}$

$y = 8$

Step 4.2.1.1.2.3

Rewrite the expression.

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

$y = 8$

Step 4.2.1.1.2.4

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

$x = 3 \cdot (2)$

$y = 8$

$x = 3 \cdot (2)$

$y = 8$

$x = 3 \cdot (2)$

$y = 8$

Step 4.2.1.2

Multiply $3$ by $2$ .

$x = 6$

$y = 8$

$x = 6$

$y = 8$

$x = 6$

$y = 8$

$x = 6$

$y = 8$

Step 5

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

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Step 5.1

Replace all occurrences of $y$ in $x = \frac{3 y}{4}$ with $- 8$ .

$x = \frac{3 (- 8)}{4}$

$y = - 8$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{3 (- 8)}{4}$ .

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Step 5.2.1.1

Cancel the common factor of $- 8$ and $4$ .

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Step 5.2.1.1.1

Factor $4$ out of $3 (- 8)$ .

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

$y = - 8$

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 $4$ .

$x = \frac{4 (3 \cdot (- 2))}{4 (1)}$

$y = - 8$

Step 5.2.1.1.2.2

Cancel the common factor.

$x = \frac{4 (3 \cdot (- 2))}{4 \cdot 1}$

$y = - 8$

Step 5.2.1.1.2.3

Rewrite the expression.

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

$y = - 8$

Step 5.2.1.1.2.4

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

$x = 3 \cdot (- 2)$

$y = - 8$

$x = 3 \cdot (- 2)$

$y = - 8$

$x = 3 \cdot (- 2)$

$y = - 8$

Step 5.2.1.2

Multiply $3$ by $- 2$ .

$x = - 6$

$y = - 8$

$x = - 6$

$y = - 8$

$x = - 6$

$y = - 8$

$x = - 6$

$y = - 8$

Step 6

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

$(6, 8)$

$(- 6, - 8)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(6, 8), (- 6, - 8)$

Equation Form:

$x = 6, y = 8$

$x = - 6, y = - 8$

Step 8