Precalculus Examples

$x + y = 36$ , $x^{2} + y^{2} = 72$

Step 1

Subtract $y$ from both sides of the equation.

$x = 36 - y$

$x^{2} + y^{2} = 72$

Step 2

Replace all occurrences of $x$ with $36 - y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x^{2} + y^{2} = 72$ with $36 - y$ .

${(36 - y)}^{2} + y^{2} = 72$

$x = 36 - y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(36 - y)}^{2} + y^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(36 - y)}^{2}$ as $(36 - y) (36 - y)$ .

$(36 - y) (36 - y) + y^{2} = 72$

$x = 36 - y$

Step 2.2.1.1.2

Expand $(36 - y) (36 - y)$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$36 (36 - y) - y (36 - y) + y^{2} = 72$

$x = 36 - y$

Step 2.2.1.1.2.2

Apply the distributive property.

$36 \cdot 36 + 36 (- y) - y (36 - y) + y^{2} = 72$

$x = 36 - y$

Step 2.2.1.1.2.3

Apply the distributive property.

$36 \cdot 36 + 36 (- y) - y \cdot 36 - y (- y) + y^{2} = 72$

$x = 36 - y$

$36 \cdot 36 + 36 (- y) - y \cdot 36 - y (- y) + y^{2} = 72$

$x = 36 - y$

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 $36$ by $36$ .

$1296 + 36 (- y) - y \cdot 36 - y (- y) + y^{2} = 72$

$x = 36 - y$

Step 2.2.1.1.3.1.2

Multiply $- 1$ by $36$ .

$1296 - 36 y - y \cdot 36 - y (- y) + y^{2} = 72$

$x = 36 - y$

Step 2.2.1.1.3.1.3

Multiply $36$ by $- 1$ .

$1296 - 36 y - 36 y - y (- y) + y^{2} = 72$

$x = 36 - y$

Step 2.2.1.1.3.1.4

Rewrite using the commutative property of multiplication.

$1296 - 36 y - 36 y - 1 \cdot (- 1 y \cdot y) + y^{2} = 72$

$x = 36 - y$

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 $y$ .

$1296 - 36 y - 36 y - 1 \cdot (- 1 (y \cdot y)) + y^{2} = 72$

$x = 36 - y$

Step 2.2.1.1.3.1.5.2

Multiply $y$ by $y$ .

$1296 - 36 y - 36 y - 1 \cdot (- 1 y^{2}) + y^{2} = 72$

$x = 36 - y$

$1296 - 36 y - 36 y - 1 \cdot (- 1 y^{2}) + y^{2} = 72$

$x = 36 - y$

Step 2.2.1.1.3.1.6

Multiply $- 1$ by $- 1$ .

$1296 - 36 y - 36 y + 1 y^{2} + y^{2} = 72$

$x = 36 - y$

Step 2.2.1.1.3.1.7

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

$1296 - 36 y - 36 y + y^{2} + y^{2} = 72$

$x = 36 - y$

$1296 - 36 y - 36 y + y^{2} + y^{2} = 72$

$x = 36 - y$

Step 2.2.1.1.3.2

Subtract $36 y$ from $- 36 y$ .

$1296 - 72 y + y^{2} + y^{2} = 72$

$x = 36 - y$

$1296 - 72 y + y^{2} + y^{2} = 72$

$x = 36 - y$

$1296 - 72 y + y^{2} + y^{2} = 72$

$x = 36 - y$

Step 2.2.1.2

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

$1296 - 72 y + 2 y^{2} = 72$

$x = 36 - y$

$1296 - 72 y + 2 y^{2} = 72$

$x = 36 - y$

$1296 - 72 y + 2 y^{2} = 72$

$x = 36 - y$

$1296 - 72 y + 2 y^{2} = 72$

$x = 36 - y$

Step 3

Solve for $y$ in $1296 - 72 y + 2 y^{2} = 72$ .

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Step 3.1

Subtract $72$ from both sides of the equation.

$1296 - 72 y + 2 y^{2} - 72 = 0$

$x = 36 - y$

Step 3.2

Subtract $72$ from $1296$ .

$- 72 y + 2 y^{2} + 1224 = 0$

$x = 36 - y$

Step 3.3

Factor the left side of the equation.

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Step 3.3.1

Factor $2$ out of $- 72 y + 2 y^{2} + 1224$ .

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Step 3.3.1.1

Factor $2$ out of $- 72 y$ .

$2 (- 36 y) + 2 y^{2} + 1224 = 0$

$x = 36 - y$

Step 3.3.1.2

Factor $2$ out of $2 y^{2}$ .

$2 (- 36 y) + 2 (y^{2}) + 1224 = 0$

$x = 36 - y$

Step 3.3.1.3

Factor $2$ out of $1224$ .

$2 (- 36 y) + 2 y^{2} + 2 \cdot 612 = 0$

$x = 36 - y$

Step 3.3.1.4

Factor $2$ out of $2 (- 36 y) + 2 y^{2}$ .

$2 (- 36 y + y^{2}) + 2 \cdot 612 = 0$

$x = 36 - y$

Step 3.3.1.5

Factor $2$ out of $2 (- 36 y + y^{2}) + 2 \cdot 612$ .

$2 (- 36 y + y^{2} + 612) = 0$

$x = 36 - y$

$2 (- 36 y + y^{2} + 612) = 0$

$x = 36 - y$

Step 3.3.2

Reorder terms.

$2 (y^{2} - 36 y + 612) = 0$

$x = 36 - y$

$2 (y^{2} - 36 y + 612) = 0$

$x = 36 - y$

Step 3.4

Divide each term in $2 (y^{2} - 36 y + 612) = 0$ by $2$ and simplify.

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Step 3.4.1

Divide each term in $2 (y^{2} - 36 y + 612) = 0$ by $2$ .

$\frac{2 (y^{2} - 36 y + 612)}{2} = \frac{0}{2}$

$x = 36 - y$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{2 (y^{2} - 36 y + 612)}{2} = \frac{0}{2}$

$x = 36 - y$

Step 3.4.2.1.2

Divide $y^{2} - 36 y + 612$ by $1$ .

$y^{2} - 36 y + 612 = \frac{0}{2}$

$x = 36 - y$

$y^{2} - 36 y + 612 = \frac{0}{2}$

$x = 36 - y$

$y^{2} - 36 y + 612 = \frac{0}{2}$

$x = 36 - y$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $0$ by $2$ .

$y^{2} - 36 y + 612 = 0$

$x = 36 - y$

$y^{2} - 36 y + 612 = 0$

$x = 36 - y$

$y^{2} - 36 y + 612 = 0$

$x = 36 - y$

Step 3.5

Use the quadratic formula to find the solutions.

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

$x = 36 - y$

Step 3.6

Substitute the values $a = 1$ , $b = - 36$ , and $c = 612$ into the quadratic formula and solve for $y$ .

$\frac{36 \pm \sqrt{{(- 36)}^{2} - 4 \cdot (1 \cdot 612)}}{2 \cdot 1}$

$x = 36 - y$

Step 3.7

Simplify.

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

Raise $- 36$ to the power of $2$ .

$y = \frac{36 \pm \sqrt{1296 - 4 \cdot 1 \cdot 612}}{2 \cdot 1}$

$x = 36 - y$

Step 3.7.1.2

Multiply $- 4 \cdot 1 \cdot 612$ .

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Step 3.7.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{36 \pm \sqrt{1296 - 4 \cdot 612}}{2 \cdot 1}$

$x = 36 - y$

Step 3.7.1.2.2

Multiply $- 4$ by $612$ .

$y = \frac{36 \pm \sqrt{1296 - 2448}}{2 \cdot 1}$

$x = 36 - y$

$y = \frac{36 \pm \sqrt{1296 - 2448}}{2 \cdot 1}$

$x = 36 - y$

Step 3.7.1.3

Subtract $2448$ from $1296$ .

$y = \frac{36 \pm \sqrt{- 1152}}{2 \cdot 1}$

$x = 36 - y$

Step 3.7.1.4

Rewrite $- 1152$ as $- 1 (1152)$ .

$y = \frac{36 \pm \sqrt{- 1 \cdot 1152}}{2 \cdot 1}$

$x = 36 - y$

Step 3.7.1.5

Rewrite $\sqrt{- 1 (1152)}$ as $\sqrt{- 1} \cdot \sqrt{1152}$ .

$y = \frac{36 \pm \sqrt{- 1} \cdot \sqrt{1152}}{2 \cdot 1}$

$x = 36 - y$

Step 3.7.1.6

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

$y = \frac{36 \pm i \cdot \sqrt{1152}}{2 \cdot 1}$

$x = 36 - y$

Step 3.7.1.7

Rewrite $1152$ as $24^{2} \cdot 2$ .

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Step 3.7.1.7.1

Factor $576$ out of $1152$ .

$y = \frac{36 \pm i \cdot \sqrt{576 (2)}}{2 \cdot 1}$

$x = 36 - y$

Step 3.7.1.7.2

Rewrite $576$ as $24^{2}$ .

$y = \frac{36 \pm i \cdot \sqrt{24^{2} \cdot 2}}{2 \cdot 1}$

$x = 36 - y$

$y = \frac{36 \pm i \cdot \sqrt{24^{2} \cdot 2}}{2 \cdot 1}$

$x = 36 - y$

Step 3.7.1.8

Pull terms out from under the radical.

$y = \frac{36 \pm i \cdot (24 \sqrt{2})}{2 \cdot 1}$

$x = 36 - y$

Step 3.7.1.9

Move $24$ to the left of $i$ .

$y = \frac{36 \pm 24 i \sqrt{2}}{2 \cdot 1}$

$x = 36 - y$

$y = \frac{36 \pm 24 i \sqrt{2}}{2 \cdot 1}$

$x = 36 - y$

Step 3.7.2

Multiply $2$ by $1$ .

$y = \frac{36 \pm 24 i \sqrt{2}}{2}$

$x = 36 - y$

Step 3.7.3

Simplify $\frac{36 \pm 24 i \sqrt{2}}{2}$ .

$y = 18 \pm 12 i \sqrt{2}$

$x = 36 - y$

$y = 18 \pm 12 i \sqrt{2}$

$x = 36 - y$

Step 3.8

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

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Step 3.8.1

Simplify the numerator.

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Step 3.8.1.1

Raise $- 36$ to the power of $2$ .

$y = \frac{36 \pm \sqrt{1296 - 4 \cdot 1 \cdot 612}}{2 \cdot 1}$

$x = 36 - y$

Step 3.8.1.2

Multiply $- 4 \cdot 1 \cdot 612$ .

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Step 3.8.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{36 \pm \sqrt{1296 - 4 \cdot 612}}{2 \cdot 1}$

$x = 36 - y$

Step 3.8.1.2.2

Multiply $- 4$ by $612$ .

$y = \frac{36 \pm \sqrt{1296 - 2448}}{2 \cdot 1}$

$x = 36 - y$

$y = \frac{36 \pm \sqrt{1296 - 2448}}{2 \cdot 1}$

$x = 36 - y$

Step 3.8.1.3

Subtract $2448$ from $1296$ .

$y = \frac{36 \pm \sqrt{- 1152}}{2 \cdot 1}$

$x = 36 - y$

Step 3.8.1.4

Rewrite $- 1152$ as $- 1 (1152)$ .

$y = \frac{36 \pm \sqrt{- 1 \cdot 1152}}{2 \cdot 1}$

$x = 36 - y$

Step 3.8.1.5

Rewrite $\sqrt{- 1 (1152)}$ as $\sqrt{- 1} \cdot \sqrt{1152}$ .

$y = \frac{36 \pm \sqrt{- 1} \cdot \sqrt{1152}}{2 \cdot 1}$

$x = 36 - y$

Step 3.8.1.6

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

$y = \frac{36 \pm i \cdot \sqrt{1152}}{2 \cdot 1}$

$x = 36 - y$

Step 3.8.1.7

Rewrite $1152$ as $24^{2} \cdot 2$ .

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Step 3.8.1.7.1

Factor $576$ out of $1152$ .

$y = \frac{36 \pm i \cdot \sqrt{576 (2)}}{2 \cdot 1}$

$x = 36 - y$

Step 3.8.1.7.2

Rewrite $576$ as $24^{2}$ .

$y = \frac{36 \pm i \cdot \sqrt{24^{2} \cdot 2}}{2 \cdot 1}$

$x = 36 - y$

$y = \frac{36 \pm i \cdot \sqrt{24^{2} \cdot 2}}{2 \cdot 1}$

$x = 36 - y$

Step 3.8.1.8

Pull terms out from under the radical.

$y = \frac{36 \pm i \cdot (24 \sqrt{2})}{2 \cdot 1}$

$x = 36 - y$

Step 3.8.1.9

Move $24$ to the left of $i$ .

$y = \frac{36 \pm 24 i \sqrt{2}}{2 \cdot 1}$

$x = 36 - y$

$y = \frac{36 \pm 24 i \sqrt{2}}{2 \cdot 1}$

$x = 36 - y$

Step 3.8.2

Multiply $2$ by $1$ .

$y = \frac{36 \pm 24 i \sqrt{2}}{2}$

$x = 36 - y$

Step 3.8.3

Simplify $\frac{36 \pm 24 i \sqrt{2}}{2}$ .

$y = 18 \pm 12 i \sqrt{2}$

$x = 36 - y$

Step 3.8.4

Change the $\pm$ to $+$ .

$y = 18 + 12 i \sqrt{2}$

$x = 36 - y$

$y = 18 + 12 i \sqrt{2}$

$x = 36 - y$

Step 3.9

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

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Step 3.9.1

Simplify the numerator.

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Step 3.9.1.1

Raise $- 36$ to the power of $2$ .

$y = \frac{36 \pm \sqrt{1296 - 4 \cdot 1 \cdot 612}}{2 \cdot 1}$

$x = 36 - y$

Step 3.9.1.2

Multiply $- 4 \cdot 1 \cdot 612$ .

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Step 3.9.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{36 \pm \sqrt{1296 - 4 \cdot 612}}{2 \cdot 1}$

$x = 36 - y$

Step 3.9.1.2.2

Multiply $- 4$ by $612$ .

$y = \frac{36 \pm \sqrt{1296 - 2448}}{2 \cdot 1}$

$x = 36 - y$

$y = \frac{36 \pm \sqrt{1296 - 2448}}{2 \cdot 1}$

$x = 36 - y$

Step 3.9.1.3

Subtract $2448$ from $1296$ .

$y = \frac{36 \pm \sqrt{- 1152}}{2 \cdot 1}$

$x = 36 - y$

Step 3.9.1.4

Rewrite $- 1152$ as $- 1 (1152)$ .

$y = \frac{36 \pm \sqrt{- 1 \cdot 1152}}{2 \cdot 1}$

$x = 36 - y$

Step 3.9.1.5

Rewrite $\sqrt{- 1 (1152)}$ as $\sqrt{- 1} \cdot \sqrt{1152}$ .

$y = \frac{36 \pm \sqrt{- 1} \cdot \sqrt{1152}}{2 \cdot 1}$

$x = 36 - y$

Step 3.9.1.6

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

$y = \frac{36 \pm i \cdot \sqrt{1152}}{2 \cdot 1}$

$x = 36 - y$

Step 3.9.1.7

Rewrite $1152$ as $24^{2} \cdot 2$ .

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Step 3.9.1.7.1

Factor $576$ out of $1152$ .

$y = \frac{36 \pm i \cdot \sqrt{576 (2)}}{2 \cdot 1}$

$x = 36 - y$

Step 3.9.1.7.2

Rewrite $576$ as $24^{2}$ .

$y = \frac{36 \pm i \cdot \sqrt{24^{2} \cdot 2}}{2 \cdot 1}$

$x = 36 - y$

$y = \frac{36 \pm i \cdot \sqrt{24^{2} \cdot 2}}{2 \cdot 1}$

$x = 36 - y$

Step 3.9.1.8

Pull terms out from under the radical.

$y = \frac{36 \pm i \cdot (24 \sqrt{2})}{2 \cdot 1}$

$x = 36 - y$

Step 3.9.1.9

Move $24$ to the left of $i$ .

$y = \frac{36 \pm 24 i \sqrt{2}}{2 \cdot 1}$

$x = 36 - y$

$y = \frac{36 \pm 24 i \sqrt{2}}{2 \cdot 1}$

$x = 36 - y$

Step 3.9.2

Multiply $2$ by $1$ .

$y = \frac{36 \pm 24 i \sqrt{2}}{2}$

$x = 36 - y$

Step 3.9.3

Simplify $\frac{36 \pm 24 i \sqrt{2}}{2}$ .

$y = 18 \pm 12 i \sqrt{2}$

$x = 36 - y$

Step 3.9.4

Change the $\pm$ to $-$ .

$y = 18 - 12 i \sqrt{2}$

$x = 36 - y$

$y = 18 - 12 i \sqrt{2}$

$x = 36 - y$

Step 3.10

The final answer is the combination of both solutions.

$y = 18 + 12 i \sqrt{2}, 18 - 12 i \sqrt{2}$

$x = 36 - y$

$y = 18 + 12 i \sqrt{2}, 18 - 12 i \sqrt{2}$

$x = 36 - y$

Step 4

Replace all occurrences of $y$ with $18 + 12 i \sqrt{2}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 36 - y$ with $18 + 12 i \sqrt{2}$ .

$x = 36 - (18 + 12 i \sqrt{2})$

$y = 18 + 12 i \sqrt{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $36 - (18 + 12 i \sqrt{2})$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$x = 36 - 1 \cdot 18 - (12 i \sqrt{2})$

$y = 18 + 12 i \sqrt{2}$

Step 4.2.1.1.2

Multiply $- 1$ by $18$ .

$x = 36 - 18 - (12 i \sqrt{2})$

$y = 18 + 12 i \sqrt{2}$

Step 4.2.1.1.3

Multiply $12$ by $- 1$ .

$x = 36 - 18 - 12 i \sqrt{2}$

$y = 18 + 12 i \sqrt{2}$

$x = 36 - 18 - 12 i \sqrt{2}$

$y = 18 + 12 i \sqrt{2}$

Step 4.2.1.2

Subtract $18$ from $36$ .

$x = 18 - 12 i \sqrt{2}$

$y = 18 + 12 i \sqrt{2}$

$x = 18 - 12 i \sqrt{2}$

$y = 18 + 12 i \sqrt{2}$

$x = 18 - 12 i \sqrt{2}$

$y = 18 + 12 i \sqrt{2}$

$x = 18 - 12 i \sqrt{2}$

$y = 18 + 12 i \sqrt{2}$

Step 5

Replace all occurrences of $y$ with $18 - 12 i \sqrt{2}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = 36 - y$ with $18 - 12 i \sqrt{2}$ .

$x = 36 - (18 - 12 i \sqrt{2})$

$y = 18 - 12 i \sqrt{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $36 - (18 - 12 i \sqrt{2})$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Apply the distributive property.

$x = 36 - 1 \cdot 18 - (- 12 i \sqrt{2})$

$y = 18 - 12 i \sqrt{2}$

Step 5.2.1.1.2

Multiply $- 1$ by $18$ .

$x = 36 - 18 - (- 12 i \sqrt{2})$

$y = 18 - 12 i \sqrt{2}$

Step 5.2.1.1.3

Multiply $- 12$ by $- 1$ .

$x = 36 - 18 + 12 i \sqrt{2}$

$y = 18 - 12 i \sqrt{2}$

$x = 36 - 18 + 12 i \sqrt{2}$

$y = 18 - 12 i \sqrt{2}$

Step 5.2.1.2

Subtract $18$ from $36$ .

$x = 18 + 12 i \sqrt{2}$

$y = 18 - 12 i \sqrt{2}$

$x = 18 + 12 i \sqrt{2}$

$y = 18 - 12 i \sqrt{2}$

$x = 18 + 12 i \sqrt{2}$

$y = 18 - 12 i \sqrt{2}$

$x = 18 + 12 i \sqrt{2}$

$y = 18 - 12 i \sqrt{2}$

Step 6

List all of the solutions.

$x = 18 - 12 i \sqrt{2}, y = 18 + 12 i \sqrt{2}$

$x = 18 + 12 i \sqrt{2}, y = 18 - 12 i \sqrt{2}$

Step 7