Precalculus Examples

$x^{2} + y^{2} = 25$ , $x + y = 8$

Step 1

Subtract $y$ from both sides of the equation.

$x = 8 - y$

$x^{2} + y^{2} = 25$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $x^{2} + y^{2} = 25$ with $8 - y$ .

${(8 - y)}^{2} + y^{2} = 25$

$x = 8 - y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(8 - y)}^{2} + y^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(8 - y)}^{2}$ as $(8 - y) (8 - y)$ .

$(8 - y) (8 - y) + y^{2} = 25$

$x = 8 - y$

Step 2.2.1.1.2

Expand $(8 - y) (8 - y)$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$8 (8 - y) - y (8 - y) + y^{2} = 25$

$x = 8 - y$

Step 2.2.1.1.2.2

Apply the distributive property.

$8 \cdot 8 + 8 (- y) - y (8 - y) + y^{2} = 25$

$x = 8 - y$

Step 2.2.1.1.2.3

Apply the distributive property.

$8 \cdot 8 + 8 (- y) - y \cdot 8 - y (- y) + y^{2} = 25$

$x = 8 - y$

$8 \cdot 8 + 8 (- y) - y \cdot 8 - y (- y) + y^{2} = 25$

$x = 8 - y$

Step 2.2.1.1.3

Simplify and combine like terms.

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Step 2.2.1.1.3.1

Simplify each term.

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Step 2.2.1.1.3.1.1

Multiply $8$ by $8$ .

$64 + 8 (- y) - y \cdot 8 - y (- y) + y^{2} = 25$

$x = 8 - y$

Step 2.2.1.1.3.1.2

Multiply $- 1$ by $8$ .

$64 - 8 y - y \cdot 8 - y (- y) + y^{2} = 25$

$x = 8 - y$

Step 2.2.1.1.3.1.3

Multiply $8$ by $- 1$ .

$64 - 8 y - 8 y - y (- y) + y^{2} = 25$

$x = 8 - y$

Step 2.2.1.1.3.1.4

Rewrite using the commutative property of multiplication.

$64 - 8 y - 8 y - 1 \cdot (- 1 y \cdot y) + y^{2} = 25$

$x = 8 - y$

Step 2.2.1.1.3.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 2.2.1.1.3.1.5.1

Move $y$ .

$64 - 8 y - 8 y - 1 \cdot (- 1 (y \cdot y)) + y^{2} = 25$

$x = 8 - y$

Step 2.2.1.1.3.1.5.2

Multiply $y$ by $y$ .

$64 - 8 y - 8 y - 1 \cdot (- 1 y^{2}) + y^{2} = 25$

$x = 8 - y$

$64 - 8 y - 8 y - 1 \cdot (- 1 y^{2}) + y^{2} = 25$

$x = 8 - y$

Step 2.2.1.1.3.1.6

Multiply $- 1$ by $- 1$ .

$64 - 8 y - 8 y + 1 y^{2} + y^{2} = 25$

$x = 8 - y$

Step 2.2.1.1.3.1.7

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

$64 - 8 y - 8 y + y^{2} + y^{2} = 25$

$x = 8 - y$

$64 - 8 y - 8 y + y^{2} + y^{2} = 25$

$x = 8 - y$

Step 2.2.1.1.3.2

Subtract $8 y$ from $- 8 y$ .

$64 - 16 y + y^{2} + y^{2} = 25$

$x = 8 - y$

$64 - 16 y + y^{2} + y^{2} = 25$

$x = 8 - y$

$64 - 16 y + y^{2} + y^{2} = 25$

$x = 8 - y$

Step 2.2.1.2

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

$64 - 16 y + 2 y^{2} = 25$

$x = 8 - y$

$64 - 16 y + 2 y^{2} = 25$

$x = 8 - y$

$64 - 16 y + 2 y^{2} = 25$

$x = 8 - y$

$64 - 16 y + 2 y^{2} = 25$

$x = 8 - y$

Step 3

Solve for $y$ in $64 - 16 y + 2 y^{2} = 25$ .

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Step 3.1

Move all terms to the left side of the equation and simplify.

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Step 3.1.1

Subtract $25$ from both sides of the equation.

$64 - 16 y + 2 y^{2} - 25 = 0$

$x = 8 - y$

Step 3.1.2

Subtract $25$ from $64$ .

$- 16 y + 2 y^{2} + 39 = 0$

$x = 8 - y$

$- 16 y + 2 y^{2} + 39 = 0$

$x = 8 - y$

Step 3.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

$x = 8 - y$

Step 3.3

Substitute the values $a = 2$ , $b = - 16$ , and $c = 39$ into the quadratic formula and solve for $y$ .

$\frac{16 \pm \sqrt{{(- 16)}^{2} - 4 \cdot (2 \cdot 39)}}{2 \cdot 2}$

$x = 8 - y$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the numerator.

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Step 3.4.1.1

Raise $- 16$ to the power of $2$ .

$y = \frac{16 \pm \sqrt{256 - 4 \cdot 2 \cdot 39}}{2 \cdot 2}$

$x = 8 - y$

Step 3.4.1.2

Multiply $- 4 \cdot 2 \cdot 39$ .

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Step 3.4.1.2.1

Multiply $- 4$ by $2$ .

$y = \frac{16 \pm \sqrt{256 - 8 \cdot 39}}{2 \cdot 2}$

$x = 8 - y$

Step 3.4.1.2.2

Multiply $- 8$ by $39$ .

$y = \frac{16 \pm \sqrt{256 - 312}}{2 \cdot 2}$

$x = 8 - y$

$y = \frac{16 \pm \sqrt{256 - 312}}{2 \cdot 2}$

$x = 8 - y$

Step 3.4.1.3

Subtract $312$ from $256$ .

$y = \frac{16 \pm \sqrt{- 56}}{2 \cdot 2}$

$x = 8 - y$

Step 3.4.1.4

Rewrite $- 56$ as $- 1 (56)$ .

$y = \frac{16 \pm \sqrt{- 1 \cdot 56}}{2 \cdot 2}$

$x = 8 - y$

Step 3.4.1.5

Rewrite $\sqrt{- 1 (56)}$ as $\sqrt{- 1} \cdot \sqrt{56}$ .

$y = \frac{16 \pm \sqrt{- 1} \cdot \sqrt{56}}{2 \cdot 2}$

$x = 8 - y$

Step 3.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{16 \pm i \cdot \sqrt{56}}{2 \cdot 2}$

$x = 8 - y$

Step 3.4.1.7

Rewrite $56$ as $2^{2} \cdot 14$ .

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Step 3.4.1.7.1

Factor $4$ out of $56$ .

$y = \frac{16 \pm i \cdot \sqrt{4 (14)}}{2 \cdot 2}$

$x = 8 - y$

Step 3.4.1.7.2

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

$y = \frac{16 \pm i \cdot \sqrt{2^{2} \cdot 14}}{2 \cdot 2}$

$x = 8 - y$

$y = \frac{16 \pm i \cdot \sqrt{2^{2} \cdot 14}}{2 \cdot 2}$

$x = 8 - y$

Step 3.4.1.8

Pull terms out from under the radical.

$y = \frac{16 \pm i \cdot (2 \sqrt{14})}{2 \cdot 2}$

$x = 8 - y$

Step 3.4.1.9

Move $2$ to the left of $i$ .

$y = \frac{16 \pm 2 i \sqrt{14}}{2 \cdot 2}$

$x = 8 - y$

$y = \frac{16 \pm 2 i \sqrt{14}}{2 \cdot 2}$

$x = 8 - y$

Step 3.4.2

Multiply $2$ by $2$ .

$y = \frac{16 \pm 2 i \sqrt{14}}{4}$

$x = 8 - y$

Step 3.4.3

Simplify $\frac{16 \pm 2 i \sqrt{14}}{4}$ .

$y = \frac{8 \pm i \sqrt{14}}{2}$

$x = 8 - y$

$y = \frac{8 \pm i \sqrt{14}}{2}$

$x = 8 - y$

Step 3.5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Raise $- 16$ to the power of $2$ .

$y = \frac{16 \pm \sqrt{256 - 4 \cdot 2 \cdot 39}}{2 \cdot 2}$

$x = 8 - y$

Step 3.5.1.2

Multiply $- 4 \cdot 2 \cdot 39$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $2$ .

$y = \frac{16 \pm \sqrt{256 - 8 \cdot 39}}{2 \cdot 2}$

$x = 8 - y$

Step 3.5.1.2.2

Multiply $- 8$ by $39$ .

$y = \frac{16 \pm \sqrt{256 - 312}}{2 \cdot 2}$

$x = 8 - y$

$y = \frac{16 \pm \sqrt{256 - 312}}{2 \cdot 2}$

$x = 8 - y$

Step 3.5.1.3

Subtract $312$ from $256$ .

$y = \frac{16 \pm \sqrt{- 56}}{2 \cdot 2}$

$x = 8 - y$

Step 3.5.1.4

Rewrite $- 56$ as $- 1 (56)$ .

$y = \frac{16 \pm \sqrt{- 1 \cdot 56}}{2 \cdot 2}$

$x = 8 - y$

Step 3.5.1.5

Rewrite $\sqrt{- 1 (56)}$ as $\sqrt{- 1} \cdot \sqrt{56}$ .

$y = \frac{16 \pm \sqrt{- 1} \cdot \sqrt{56}}{2 \cdot 2}$

$x = 8 - y$

Step 3.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{16 \pm i \cdot \sqrt{56}}{2 \cdot 2}$

$x = 8 - y$

Step 3.5.1.7

Rewrite $56$ as $2^{2} \cdot 14$ .

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Step 3.5.1.7.1

Factor $4$ out of $56$ .

$y = \frac{16 \pm i \cdot \sqrt{4 (14)}}{2 \cdot 2}$

$x = 8 - y$

Step 3.5.1.7.2

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

$y = \frac{16 \pm i \cdot \sqrt{2^{2} \cdot 14}}{2 \cdot 2}$

$x = 8 - y$

$y = \frac{16 \pm i \cdot \sqrt{2^{2} \cdot 14}}{2 \cdot 2}$

$x = 8 - y$

Step 3.5.1.8

Pull terms out from under the radical.

$y = \frac{16 \pm i \cdot (2 \sqrt{14})}{2 \cdot 2}$

$x = 8 - y$

Step 3.5.1.9

Move $2$ to the left of $i$ .

$y = \frac{16 \pm 2 i \sqrt{14}}{2 \cdot 2}$

$x = 8 - y$

$y = \frac{16 \pm 2 i \sqrt{14}}{2 \cdot 2}$

$x = 8 - y$

Step 3.5.2

Multiply $2$ by $2$ .

$y = \frac{16 \pm 2 i \sqrt{14}}{4}$

$x = 8 - y$

Step 3.5.3

Simplify $\frac{16 \pm 2 i \sqrt{14}}{4}$ .

$y = \frac{8 \pm i \sqrt{14}}{2}$

$x = 8 - y$

Step 3.5.4

Change the $\pm$ to $+$ .

$y = \frac{8 + i \sqrt{14}}{2}$

$x = 8 - y$

$y = \frac{8 + i \sqrt{14}}{2}$

$x = 8 - y$

Step 3.6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $- 16$ to the power of $2$ .

$y = \frac{16 \pm \sqrt{256 - 4 \cdot 2 \cdot 39}}{2 \cdot 2}$

$x = 8 - y$

Step 3.6.1.2

Multiply $- 4 \cdot 2 \cdot 39$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $2$ .

$y = \frac{16 \pm \sqrt{256 - 8 \cdot 39}}{2 \cdot 2}$

$x = 8 - y$

Step 3.6.1.2.2

Multiply $- 8$ by $39$ .

$y = \frac{16 \pm \sqrt{256 - 312}}{2 \cdot 2}$

$x = 8 - y$

$y = \frac{16 \pm \sqrt{256 - 312}}{2 \cdot 2}$

$x = 8 - y$

Step 3.6.1.3

Subtract $312$ from $256$ .

$y = \frac{16 \pm \sqrt{- 56}}{2 \cdot 2}$

$x = 8 - y$

Step 3.6.1.4

Rewrite $- 56$ as $- 1 (56)$ .

$y = \frac{16 \pm \sqrt{- 1 \cdot 56}}{2 \cdot 2}$

$x = 8 - y$

Step 3.6.1.5

Rewrite $\sqrt{- 1 (56)}$ as $\sqrt{- 1} \cdot \sqrt{56}$ .

$y = \frac{16 \pm \sqrt{- 1} \cdot \sqrt{56}}{2 \cdot 2}$

$x = 8 - y$

Step 3.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{16 \pm i \cdot \sqrt{56}}{2 \cdot 2}$

$x = 8 - y$

Step 3.6.1.7

Rewrite $56$ as $2^{2} \cdot 14$ .

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Step 3.6.1.7.1

Factor $4$ out of $56$ .

$y = \frac{16 \pm i \cdot \sqrt{4 (14)}}{2 \cdot 2}$

$x = 8 - y$

Step 3.6.1.7.2

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

$y = \frac{16 \pm i \cdot \sqrt{2^{2} \cdot 14}}{2 \cdot 2}$

$x = 8 - y$

$y = \frac{16 \pm i \cdot \sqrt{2^{2} \cdot 14}}{2 \cdot 2}$

$x = 8 - y$

Step 3.6.1.8

Pull terms out from under the radical.

$y = \frac{16 \pm i \cdot (2 \sqrt{14})}{2 \cdot 2}$

$x = 8 - y$

Step 3.6.1.9

Move $2$ to the left of $i$ .

$y = \frac{16 \pm 2 i \sqrt{14}}{2 \cdot 2}$

$x = 8 - y$

$y = \frac{16 \pm 2 i \sqrt{14}}{2 \cdot 2}$

$x = 8 - y$

Step 3.6.2

Multiply $2$ by $2$ .

$y = \frac{16 \pm 2 i \sqrt{14}}{4}$

$x = 8 - y$

Step 3.6.3

Simplify $\frac{16 \pm 2 i \sqrt{14}}{4}$ .

$y = \frac{8 \pm i \sqrt{14}}{2}$

$x = 8 - y$

Step 3.6.4

Change the $\pm$ to $-$ .

$y = \frac{8 - i \sqrt{14}}{2}$

$x = 8 - y$

$y = \frac{8 - i \sqrt{14}}{2}$

$x = 8 - y$

Step 3.7

The final answer is the combination of both solutions.

$y = \frac{8 + i \sqrt{14}}{2}, \frac{8 - i \sqrt{14}}{2}$

$x = 8 - y$

$y = \frac{8 + i \sqrt{14}}{2}, \frac{8 - i \sqrt{14}}{2}$

$x = 8 - y$

Step 4

Replace all occurrences of $y$ with $\frac{8 + i \sqrt{14}}{2}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 8 - y$ with $\frac{8 + i \sqrt{14}}{2}$ .

$x = 8 - (\frac{8 + i \sqrt{14}}{2})$

$y = \frac{8 + i \sqrt{14}}{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $8 - (\frac{8 + i \sqrt{14}}{2})$ .

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Step 4.2.1.1

To write $8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = 8 \cdot \frac{2}{2} - \frac{8 + i \sqrt{14}}{2}$

$y = \frac{8 + i \sqrt{14}}{2}$

Step 4.2.1.2

Combine $8$ and $\frac{2}{2}$ .

$x = \frac{8 \cdot 2}{2} - \frac{8 + i \sqrt{14}}{2}$

$y = \frac{8 + i \sqrt{14}}{2}$

Step 4.2.1.3

Combine the numerators over the common denominator.

$x = \frac{8 - i \sqrt{14}}{2}$

$y = \frac{8 + i \sqrt{14}}{2}$

$x = \frac{8 - i \sqrt{14}}{2}$

$y = \frac{8 + i \sqrt{14}}{2}$

$x = \frac{8 - i \sqrt{14}}{2}$

$y = \frac{8 + i \sqrt{14}}{2}$

$x = \frac{8 - i \sqrt{14}}{2}$

$y = \frac{8 + i \sqrt{14}}{2}$

Step 5

Replace all occurrences of $y$ with $\frac{8 - i \sqrt{14}}{2}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = 8 - y$ with $\frac{8 - i \sqrt{14}}{2}$ .

$x = 8 - (\frac{8 - i \sqrt{14}}{2})$

$y = \frac{8 - i \sqrt{14}}{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $8 - (\frac{8 - i \sqrt{14}}{2})$ .

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Step 5.2.1.1

To write $8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = 8 \cdot \frac{2}{2} - \frac{8 - i \sqrt{14}}{2}$

$y = \frac{8 - i \sqrt{14}}{2}$

Step 5.2.1.2

Combine $8$ and $\frac{2}{2}$ .

$x = \frac{8 \cdot 2}{2} - \frac{8 - i \sqrt{14}}{2}$

$y = \frac{8 - i \sqrt{14}}{2}$

Step 5.2.1.3

Combine the numerators over the common denominator.

$x = \frac{8 + \sqrt{14} i}{2}$

$y = \frac{8 - i \sqrt{14}}{2}$

$x = \frac{8 + \sqrt{14} i}{2}$

$y = \frac{8 - i \sqrt{14}}{2}$

$x = \frac{8 + \sqrt{14} i}{2}$

$y = \frac{8 - i \sqrt{14}}{2}$

$x = \frac{8 + \sqrt{14} i}{2}$

$y = \frac{8 - i \sqrt{14}}{2}$

Step 6

List all of the solutions.

$x = \frac{8 - i \sqrt{14}}{2}, y = \frac{8 + i \sqrt{14}}{2}$

$x = \frac{8 + \sqrt{14} i}{2}, y = \frac{8 - i \sqrt{14}}{2}$

Step 7