Precalculus Examples

$y - x = 5$ , $8 x^{2} - 13 x - y = 9$

Step 1

Add $x$ to both sides of the equation.

$y = 5 + x$

$8 x^{2} - 13 x - y = 9$

Step 2

Replace all occurrences of $y$ with $5 + x$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $8 x^{2} - 13 x - y = 9$ with $5 + x$ .

$8 x^{2} - 13 x - (5 + x) = 9$

$y = 5 + x$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $8 x^{2} - 13 x - (5 + x)$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$8 x^{2} - 13 x - 1 \cdot 5 - x = 9$

$y = 5 + x$

Step 2.2.1.1.2

Multiply $- 1$ by $5$ .

$8 x^{2} - 13 x - 5 - x = 9$

$y = 5 + x$

$8 x^{2} - 13 x - 5 - x = 9$

$y = 5 + x$

Step 2.2.1.2

Subtract $x$ from $- 13 x$ .

$8 x^{2} - 14 x - 5 = 9$

$y = 5 + x$

$8 x^{2} - 14 x - 5 = 9$

$y = 5 + x$

$8 x^{2} - 14 x - 5 = 9$

$y = 5 + x$

$8 x^{2} - 14 x - 5 = 9$

$y = 5 + x$

Step 3

Solve for $x$ in $8 x^{2} - 14 x - 5 = 9$ .

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Step 3.1

Subtract $9$ from both sides of the equation.

$8 x^{2} - 14 x - 5 - 9 = 0$

$y = 5 + x$

Step 3.2

Subtract $9$ from $- 5$ .

$8 x^{2} - 14 x - 14 = 0$

$y = 5 + x$

Step 3.3

Factor $2$ out of $8 x^{2} - 14 x - 14$ .

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Step 3.3.1

Factor $2$ out of $8 x^{2}$ .

$2 (4 x^{2}) - 14 x - 14 = 0$

$y = 5 + x$

Step 3.3.2

Factor $2$ out of $- 14 x$ .

$2 (4 x^{2}) + 2 (- 7 x) - 14 = 0$

$y = 5 + x$

Step 3.3.3

Factor $2$ out of $- 14$ .

$2 (4 x^{2}) + 2 (- 7 x) + 2 (- 7) = 0$

$y = 5 + x$

Step 3.3.4

Factor $2$ out of $2 (4 x^{2}) + 2 (- 7 x)$ .

$2 (4 x^{2} - 7 x) + 2 (- 7) = 0$

$y = 5 + x$

Step 3.3.5

Factor $2$ out of $2 (4 x^{2} - 7 x) + 2 (- 7)$ .

$2 (4 x^{2} - 7 x - 7) = 0$

$y = 5 + x$

$2 (4 x^{2} - 7 x - 7) = 0$

$y = 5 + x$

Step 3.4

Divide each term in $2 (4 x^{2} - 7 x - 7) = 0$ by $2$ and simplify.

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Step 3.4.1

Divide each term in $2 (4 x^{2} - 7 x - 7) = 0$ by $2$ .

$\frac{2 (4 x^{2} - 7 x - 7)}{2} = \frac{0}{2}$

$y = 5 + x$

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.

$\frac{2 (4 x^{2} - 7 x - 7)}{2} = \frac{0}{2}$

$y = 5 + x$

Step 3.4.2.1.2

Divide $4 x^{2} - 7 x - 7$ by $1$ .

$4 x^{2} - 7 x - 7 = \frac{0}{2}$

$y = 5 + x$

$4 x^{2} - 7 x - 7 = \frac{0}{2}$

$y = 5 + x$

$4 x^{2} - 7 x - 7 = \frac{0}{2}$

$y = 5 + x$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $0$ by $2$ .

$4 x^{2} - 7 x - 7 = 0$

$y = 5 + x$

$4 x^{2} - 7 x - 7 = 0$

$y = 5 + x$

$4 x^{2} - 7 x - 7 = 0$

$y = 5 + x$

Step 3.5

Use the quadratic formula to find the solutions.

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

$y = 5 + x$

Step 3.6

Substitute the values $a = 4$ , $b = - 7$ , and $c = - 7$ into the quadratic formula and solve for $x$ .

$\frac{7 \pm \sqrt{{(- 7)}^{2} - 4 \cdot (4 \cdot - 7)}}{2 \cdot 4}$

$y = 5 + x$

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 $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 4 \cdot - 7}}{2 \cdot 4}$

$y = 5 + x$

Step 3.7.1.2

Multiply $- 4 \cdot 4 \cdot - 7$ .

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Step 3.7.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{7 \pm \sqrt{49 - 16 \cdot - 7}}{2 \cdot 4}$

$y = 5 + x$

Step 3.7.1.2.2

Multiply $- 16$ by $- 7$ .

$x = \frac{7 \pm \sqrt{49 + 112}}{2 \cdot 4}$

$y = 5 + x$

$x = \frac{7 \pm \sqrt{49 + 112}}{2 \cdot 4}$

$y = 5 + x$

Step 3.7.1.3

Add $49$ and $112$ .

$x = \frac{7 \pm \sqrt{161}}{2 \cdot 4}$

$y = 5 + x$

$x = \frac{7 \pm \sqrt{161}}{2 \cdot 4}$

$y = 5 + x$

Step 3.7.2

Multiply $2$ by $4$ .

$x = \frac{7 \pm \sqrt{161}}{8}$

$y = 5 + x$

$x = \frac{7 \pm \sqrt{161}}{8}$

$y = 5 + x$

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 $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 4 \cdot - 7}}{2 \cdot 4}$

$y = 5 + x$

Step 3.8.1.2

Multiply $- 4 \cdot 4 \cdot - 7$ .

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Step 3.8.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{7 \pm \sqrt{49 - 16 \cdot - 7}}{2 \cdot 4}$

$y = 5 + x$

Step 3.8.1.2.2

Multiply $- 16$ by $- 7$ .

$x = \frac{7 \pm \sqrt{49 + 112}}{2 \cdot 4}$

$y = 5 + x$

$x = \frac{7 \pm \sqrt{49 + 112}}{2 \cdot 4}$

$y = 5 + x$

Step 3.8.1.3

Add $49$ and $112$ .

$x = \frac{7 \pm \sqrt{161}}{2 \cdot 4}$

$y = 5 + x$

$x = \frac{7 \pm \sqrt{161}}{2 \cdot 4}$

$y = 5 + x$

Step 3.8.2

Multiply $2$ by $4$ .

$x = \frac{7 \pm \sqrt{161}}{8}$

$y = 5 + x$

Step 3.8.3

Change the $\pm$ to $+$ .

$x = \frac{7 + \sqrt{161}}{8}$

$y = 5 + x$

$x = \frac{7 + \sqrt{161}}{8}$

$y = 5 + x$

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 $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 4 \cdot - 7}}{2 \cdot 4}$

$y = 5 + x$

Step 3.9.1.2

Multiply $- 4 \cdot 4 \cdot - 7$ .

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Step 3.9.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{7 \pm \sqrt{49 - 16 \cdot - 7}}{2 \cdot 4}$

$y = 5 + x$

Step 3.9.1.2.2

Multiply $- 16$ by $- 7$ .

$x = \frac{7 \pm \sqrt{49 + 112}}{2 \cdot 4}$

$y = 5 + x$

$x = \frac{7 \pm \sqrt{49 + 112}}{2 \cdot 4}$

$y = 5 + x$

Step 3.9.1.3

Add $49$ and $112$ .

$x = \frac{7 \pm \sqrt{161}}{2 \cdot 4}$

$y = 5 + x$

$x = \frac{7 \pm \sqrt{161}}{2 \cdot 4}$

$y = 5 + x$

Step 3.9.2

Multiply $2$ by $4$ .

$x = \frac{7 \pm \sqrt{161}}{8}$

$y = 5 + x$

Step 3.9.3

Change the $\pm$ to $-$ .

$x = \frac{7 - \sqrt{161}}{8}$

$y = 5 + x$

$x = \frac{7 - \sqrt{161}}{8}$

$y = 5 + x$

Step 3.10

The final answer is the combination of both solutions.

$x = \frac{7 + \sqrt{161}}{8}, \frac{7 - \sqrt{161}}{8}$

$y = 5 + x$

$x = \frac{7 + \sqrt{161}}{8}, \frac{7 - \sqrt{161}}{8}$

$y = 5 + x$

Step 4

Replace all occurrences of $x$ with $\frac{7 + \sqrt{161}}{8}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = 5 + x$ with $\frac{7 + \sqrt{161}}{8}$ .

$y = 5 + \frac{7 + \sqrt{161}}{8}$

$x = \frac{7 + \sqrt{161}}{8}$

Step 4.2

Simplify $y = 5 + \frac{7 + \sqrt{161}}{8}$ .

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Remove parentheses.

$y = 5 + \frac{7 + \sqrt{161}}{8}$

$x = \frac{7 + \sqrt{161}}{8}$

$y = 5 + \frac{7 + \sqrt{161}}{8}$

$x = \frac{7 + \sqrt{161}}{8}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $5 + \frac{7 + \sqrt{161}}{8}$ .

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Step 4.2.2.1.1

To write $5$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y = 5 \cdot \frac{8}{8} + \frac{7 + \sqrt{161}}{8}$

$x = \frac{7 + \sqrt{161}}{8}$

Step 4.2.2.1.2

Combine fractions.

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Step 4.2.2.1.2.1

Combine $5$ and $\frac{8}{8}$ .

$y = \frac{5 \cdot 8}{8} + \frac{7 + \sqrt{161}}{8}$

$x = \frac{7 + \sqrt{161}}{8}$

Step 4.2.2.1.2.2

Combine the numerators over the common denominator.

$y = \frac{5 \cdot 8 + 7 + \sqrt{161}}{8}$

$x = \frac{7 + \sqrt{161}}{8}$

$y = \frac{5 \cdot 8 + 7 + \sqrt{161}}{8}$

$x = \frac{7 + \sqrt{161}}{8}$

Step 4.2.2.1.3

Simplify the numerator.

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Step 4.2.2.1.3.1

Multiply $5$ by $8$ .

$y = \frac{40 + 7 + \sqrt{161}}{8}$

$x = \frac{7 + \sqrt{161}}{8}$

Step 4.2.2.1.3.2

Add $40$ and $7$ .

$y = \frac{47 + \sqrt{161}}{8}$

$x = \frac{7 + \sqrt{161}}{8}$

$y = \frac{47 + \sqrt{161}}{8}$

$x = \frac{7 + \sqrt{161}}{8}$

$y = \frac{47 + \sqrt{161}}{8}$

$x = \frac{7 + \sqrt{161}}{8}$

$y = \frac{47 + \sqrt{161}}{8}$

$x = \frac{7 + \sqrt{161}}{8}$

$y = \frac{47 + \sqrt{161}}{8}$

$x = \frac{7 + \sqrt{161}}{8}$

$y = \frac{47 + \sqrt{161}}{8}$

$x = \frac{7 + \sqrt{161}}{8}$

Step 5

Replace all occurrences of $x$ with $\frac{7 - \sqrt{161}}{8}$ in each equation.

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Step 5.1

Replace all occurrences of $x$ in $y = 5 + x$ with $\frac{7 - \sqrt{161}}{8}$ .

$y = 5 + \frac{7 - \sqrt{161}}{8}$

$x = \frac{7 - \sqrt{161}}{8}$

Step 5.2

Simplify $y = 5 + \frac{7 - \sqrt{161}}{8}$ .

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Remove parentheses.

$y = 5 + \frac{7 - \sqrt{161}}{8}$

$x = \frac{7 - \sqrt{161}}{8}$

$y = 5 + \frac{7 - \sqrt{161}}{8}$

$x = \frac{7 - \sqrt{161}}{8}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $5 + \frac{7 - \sqrt{161}}{8}$ .

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Step 5.2.2.1.1

To write $5$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y = 5 \cdot \frac{8}{8} + \frac{7 - \sqrt{161}}{8}$

$x = \frac{7 - \sqrt{161}}{8}$

Step 5.2.2.1.2

Combine fractions.

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Step 5.2.2.1.2.1

Combine $5$ and $\frac{8}{8}$ .

$y = \frac{5 \cdot 8}{8} + \frac{7 - \sqrt{161}}{8}$

$x = \frac{7 - \sqrt{161}}{8}$

Step 5.2.2.1.2.2

Combine the numerators over the common denominator.

$y = \frac{5 \cdot 8 + 7 - \sqrt{161}}{8}$

$x = \frac{7 - \sqrt{161}}{8}$

$y = \frac{5 \cdot 8 + 7 - \sqrt{161}}{8}$

$x = \frac{7 - \sqrt{161}}{8}$

Step 5.2.2.1.3

Simplify the numerator.

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Step 5.2.2.1.3.1

Multiply $5$ by $8$ .

$y = \frac{40 + 7 - \sqrt{161}}{8}$

$x = \frac{7 - \sqrt{161}}{8}$

Step 5.2.2.1.3.2

Add $40$ and $7$ .

$y = \frac{47 - \sqrt{161}}{8}$

$x = \frac{7 - \sqrt{161}}{8}$

$y = \frac{47 - \sqrt{161}}{8}$

$x = \frac{7 - \sqrt{161}}{8}$

$y = \frac{47 - \sqrt{161}}{8}$

$x = \frac{7 - \sqrt{161}}{8}$

$y = \frac{47 - \sqrt{161}}{8}$

$x = \frac{7 - \sqrt{161}}{8}$

$y = \frac{47 - \sqrt{161}}{8}$

$x = \frac{7 - \sqrt{161}}{8}$

$y = \frac{47 - \sqrt{161}}{8}$

$x = \frac{7 - \sqrt{161}}{8}$

Step 6

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

$(\frac{7 + \sqrt{161}}{8}, \frac{47 + \sqrt{161}}{8})$

$(\frac{7 - \sqrt{161}}{8}, \frac{47 - \sqrt{161}}{8})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(\frac{7 + \sqrt{161}}{8}, \frac{47 + \sqrt{161}}{8}), (\frac{7 - \sqrt{161}}{8}, \frac{47 - \sqrt{161}}{8})$

Equation Form:

$x = \frac{7 + \sqrt{161}}{8}, y = \frac{47 + \sqrt{161}}{8}$

$x = \frac{7 - \sqrt{161}}{8}, y = \frac{47 - \sqrt{161}}{8}$

Step 8