Precalculus Examples

$x - y = - 4$ , $x^{2} - y = - 6$

Step 1

Add $y$ to both sides of the equation.

$x = - 4 + y$

$x^{2} - y = - 6$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $x^{2} - y = - 6$ with $- 4 + y$ .

${(- 4 + y)}^{2} - y = - 6$

$x = - 4 + y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(- 4 + y)}^{2} - y$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(- 4 + y)}^{2}$ as $(- 4 + y) (- 4 + y)$ .

$(- 4 + y) (- 4 + y) - y = - 6$

$x = - 4 + y$

Step 2.2.1.1.2

Expand $(- 4 + y) (- 4 + y)$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$- 4 (- 4 + y) + y (- 4 + y) - y = - 6$

$x = - 4 + y$

Step 2.2.1.1.2.2

Apply the distributive property.

$- 4 \cdot - 4 - 4 y + y (- 4 + y) - y = - 6$

$x = - 4 + y$

Step 2.2.1.1.2.3

Apply the distributive property.

$- 4 \cdot - 4 - 4 y + y \cdot - 4 + y \cdot y - y = - 6$

$x = - 4 + y$

$- 4 \cdot - 4 - 4 y + y \cdot - 4 + y \cdot y - y = - 6$

$x = - 4 + y$

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 $- 4$ by $- 4$ .

$16 - 4 y + y \cdot - 4 + y \cdot y - y = - 6$

$x = - 4 + y$

Step 2.2.1.1.3.1.2

Move $- 4$ to the left of $y$ .

$16 - 4 y - 4 \cdot y + y \cdot y - y = - 6$

$x = - 4 + y$

Step 2.2.1.1.3.1.3

Multiply $y$ by $y$ .

$16 - 4 y - 4 y + y^{2} - y = - 6$

$x = - 4 + y$

$16 - 4 y - 4 y + y^{2} - y = - 6$

$x = - 4 + y$

Step 2.2.1.1.3.2

Subtract $4 y$ from $- 4 y$ .

$16 - 8 y + y^{2} - y = - 6$

$x = - 4 + y$

$16 - 8 y + y^{2} - y = - 6$

$x = - 4 + y$

$16 - 8 y + y^{2} - y = - 6$

$x = - 4 + y$

Step 2.2.1.2

Subtract $y$ from $- 8 y$ .

$16 + y^{2} - 9 y = - 6$

$x = - 4 + y$

$16 + y^{2} - 9 y = - 6$

$x = - 4 + y$

$16 + y^{2} - 9 y = - 6$

$x = - 4 + y$

$16 + y^{2} - 9 y = - 6$

$x = - 4 + y$

Step 3

Solve for $y$ in $16 + y^{2} - 9 y = - 6$ .

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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

Add $6$ to both sides of the equation.

$16 + y^{2} - 9 y + 6 = 0$

$x = - 4 + y$

Step 3.1.2

Add $16$ and $6$ .

$y^{2} - 9 y + 22 = 0$

$x = - 4 + y$

$y^{2} - 9 y + 22 = 0$

$x = - 4 + y$

Step 3.2

Use the quadratic formula to find the solutions.

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

$x = - 4 + y$

Step 3.3

Substitute the values $a = 1$ , $b = - 9$ , and $c = 22$ into the quadratic formula and solve for $y$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (1 \cdot 22)}}{2 \cdot 1}$

$x = - 4 + y$

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 $- 9$ to the power of $2$ .

$y = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot 22}}{2 \cdot 1}$

$x = - 4 + y$

Step 3.4.1.2

Multiply $- 4 \cdot 1 \cdot 22$ .

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Step 3.4.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{9 \pm \sqrt{81 - 4 \cdot 22}}{2 \cdot 1}$

$x = - 4 + y$

Step 3.4.1.2.2

Multiply $- 4$ by $22$ .

$y = \frac{9 \pm \sqrt{81 - 88}}{2 \cdot 1}$

$x = - 4 + y$

$y = \frac{9 \pm \sqrt{81 - 88}}{2 \cdot 1}$

$x = - 4 + y$

Step 3.4.1.3

Subtract $88$ from $81$ .

$y = \frac{9 \pm \sqrt{- 7}}{2 \cdot 1}$

$x = - 4 + y$

Step 3.4.1.4

Rewrite $- 7$ as $- 1 (7)$ .

$y = \frac{9 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 1}$

$x = - 4 + y$

Step 3.4.1.5

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$y = \frac{9 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 1}$

$x = - 4 + y$

Step 3.4.1.6

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

$y = \frac{9 \pm i \sqrt{7}}{2 \cdot 1}$

$x = - 4 + y$

$y = \frac{9 \pm i \sqrt{7}}{2 \cdot 1}$

$x = - 4 + y$

Step 3.4.2

Multiply $2$ by $1$ .

$y = \frac{9 \pm i \sqrt{7}}{2}$

$x = - 4 + y$

$y = \frac{9 \pm i \sqrt{7}}{2}$

$x = - 4 + y$

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 $- 9$ to the power of $2$ .

$y = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot 22}}{2 \cdot 1}$

$x = - 4 + y$

Step 3.5.1.2

Multiply $- 4 \cdot 1 \cdot 22$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{9 \pm \sqrt{81 - 4 \cdot 22}}{2 \cdot 1}$

$x = - 4 + y$

Step 3.5.1.2.2

Multiply $- 4$ by $22$ .

$y = \frac{9 \pm \sqrt{81 - 88}}{2 \cdot 1}$

$x = - 4 + y$

$y = \frac{9 \pm \sqrt{81 - 88}}{2 \cdot 1}$

$x = - 4 + y$

Step 3.5.1.3

Subtract $88$ from $81$ .

$y = \frac{9 \pm \sqrt{- 7}}{2 \cdot 1}$

$x = - 4 + y$

Step 3.5.1.4

Rewrite $- 7$ as $- 1 (7)$ .

$y = \frac{9 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 1}$

$x = - 4 + y$

Step 3.5.1.5

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$y = \frac{9 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 1}$

$x = - 4 + y$

Step 3.5.1.6

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

$y = \frac{9 \pm i \sqrt{7}}{2 \cdot 1}$

$x = - 4 + y$

$y = \frac{9 \pm i \sqrt{7}}{2 \cdot 1}$

$x = - 4 + y$

Step 3.5.2

Multiply $2$ by $1$ .

$y = \frac{9 \pm i \sqrt{7}}{2}$

$x = - 4 + y$

Step 3.5.3

Change the $\pm$ to $+$ .

$y = \frac{9 + i \sqrt{7}}{2}$

$x = - 4 + y$

$y = \frac{9 + i \sqrt{7}}{2}$

$x = - 4 + y$

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 $- 9$ to the power of $2$ .

$y = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot 22}}{2 \cdot 1}$

$x = - 4 + y$

Step 3.6.1.2

Multiply $- 4 \cdot 1 \cdot 22$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{9 \pm \sqrt{81 - 4 \cdot 22}}{2 \cdot 1}$

$x = - 4 + y$

Step 3.6.1.2.2

Multiply $- 4$ by $22$ .

$y = \frac{9 \pm \sqrt{81 - 88}}{2 \cdot 1}$

$x = - 4 + y$

$y = \frac{9 \pm \sqrt{81 - 88}}{2 \cdot 1}$

$x = - 4 + y$

Step 3.6.1.3

Subtract $88$ from $81$ .

$y = \frac{9 \pm \sqrt{- 7}}{2 \cdot 1}$

$x = - 4 + y$

Step 3.6.1.4

Rewrite $- 7$ as $- 1 (7)$ .

$y = \frac{9 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 1}$

$x = - 4 + y$

Step 3.6.1.5

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$y = \frac{9 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 1}$

$x = - 4 + y$

Step 3.6.1.6

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

$y = \frac{9 \pm i \sqrt{7}}{2 \cdot 1}$

$x = - 4 + y$

$y = \frac{9 \pm i \sqrt{7}}{2 \cdot 1}$

$x = - 4 + y$

Step 3.6.2

Multiply $2$ by $1$ .

$y = \frac{9 \pm i \sqrt{7}}{2}$

$x = - 4 + y$

Step 3.6.3

Change the $\pm$ to $-$ .

$y = \frac{9 - i \sqrt{7}}{2}$

$x = - 4 + y$

$y = \frac{9 - i \sqrt{7}}{2}$

$x = - 4 + y$

Step 3.7

The final answer is the combination of both solutions.

$y = \frac{9 + i \sqrt{7}}{2}, \frac{9 - i \sqrt{7}}{2}$

$x = - 4 + y$

$y = \frac{9 + i \sqrt{7}}{2}, \frac{9 - i \sqrt{7}}{2}$

$x = - 4 + y$

Step 4

Replace all occurrences of $y$ with $\frac{9 + i \sqrt{7}}{2}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - 4 + y$ with $\frac{9 + i \sqrt{7}}{2}$ .

$x = - 4 + \frac{9 + i \sqrt{7}}{2}$

$y = \frac{9 + i \sqrt{7}}{2}$

Step 4.2

Simplify $x = - 4 + \frac{9 + i \sqrt{7}}{2}$ .

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Remove parentheses.

$x = - 4 + \frac{9 + i \sqrt{7}}{2}$

$y = \frac{9 + i \sqrt{7}}{2}$

$x = - 4 + \frac{9 + i \sqrt{7}}{2}$

$y = \frac{9 + i \sqrt{7}}{2}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $- 4 + \frac{9 + i \sqrt{7}}{2}$ .

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Step 4.2.2.1.1

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

$x = - 4 \cdot \frac{2}{2} + \frac{9 + i \sqrt{7}}{2}$

$y = \frac{9 + i \sqrt{7}}{2}$

Step 4.2.2.1.2

Combine $- 4$ and $\frac{2}{2}$ .

$x = \frac{- 4 \cdot 2}{2} + \frac{9 + i \sqrt{7}}{2}$

$y = \frac{9 + i \sqrt{7}}{2}$

Step 4.2.2.1.3

Combine the numerators over the common denominator.

$x = \frac{1 + i \sqrt{7}}{2}$

$y = \frac{9 + i \sqrt{7}}{2}$

$x = \frac{1 + i \sqrt{7}}{2}$

$y = \frac{9 + i \sqrt{7}}{2}$

$x = \frac{1 + i \sqrt{7}}{2}$

$y = \frac{9 + i \sqrt{7}}{2}$

$x = \frac{1 + i \sqrt{7}}{2}$

$y = \frac{9 + i \sqrt{7}}{2}$

$x = \frac{1 + i \sqrt{7}}{2}$

$y = \frac{9 + i \sqrt{7}}{2}$

Step 5

Replace all occurrences of $y$ with $\frac{9 - i \sqrt{7}}{2}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = - 4 + y$ with $\frac{9 - i \sqrt{7}}{2}$ .

$x = - 4 + \frac{9 - i \sqrt{7}}{2}$

$y = \frac{9 - i \sqrt{7}}{2}$

Step 5.2

Simplify $x = - 4 + \frac{9 - i \sqrt{7}}{2}$ .

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Remove parentheses.

$x = - 4 + \frac{9 - i \sqrt{7}}{2}$

$y = \frac{9 - i \sqrt{7}}{2}$

$x = - 4 + \frac{9 - i \sqrt{7}}{2}$

$y = \frac{9 - i \sqrt{7}}{2}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $- 4 + \frac{9 - i \sqrt{7}}{2}$ .

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Step 5.2.2.1.1

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

$x = - 4 \cdot \frac{2}{2} + \frac{9 - i \sqrt{7}}{2}$

$y = \frac{9 - i \sqrt{7}}{2}$

Step 5.2.2.1.2

Combine $- 4$ and $\frac{2}{2}$ .

$x = \frac{- 4 \cdot 2}{2} + \frac{9 - i \sqrt{7}}{2}$

$y = \frac{9 - i \sqrt{7}}{2}$

Step 5.2.2.1.3

Combine the numerators over the common denominator.

$x = \frac{1 - i \sqrt{7}}{2}$

$y = \frac{9 - i \sqrt{7}}{2}$

$x = \frac{1 - i \sqrt{7}}{2}$

$y = \frac{9 - i \sqrt{7}}{2}$

$x = \frac{1 - i \sqrt{7}}{2}$

$y = \frac{9 - i \sqrt{7}}{2}$

$x = \frac{1 - i \sqrt{7}}{2}$

$y = \frac{9 - i \sqrt{7}}{2}$

$x = \frac{1 - i \sqrt{7}}{2}$

$y = \frac{9 - i \sqrt{7}}{2}$

Step 6

List all of the solutions.

$x = \frac{1 + i \sqrt{7}}{2}, y = \frac{9 + i \sqrt{7}}{2}$

$x = \frac{1 - i \sqrt{7}}{2}, y = \frac{9 - i \sqrt{7}}{2}$

Step 7