Precalculus Examples

$x^{2} + y^{2} = 5$ , $y = \frac{2}{3} x + 1$

Step 1

Combine $\frac{2}{3}$ and $x$ .

$y = \frac{2 x}{3} + 1$

$x^{2} + y^{2} = 5$

Step 2

Replace all occurrences of $y$ with $\frac{2 x}{3} + 1$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $x^{2} + y^{2} = 5$ with $\frac{2 x}{3} + 1$ .

$x^{2} + {(\frac{2 x}{3} + 1)}^{2} = 5$

$y = \frac{2 x}{3} + 1$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $x^{2} + {(\frac{2 x}{3} + 1)}^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(\frac{2 x}{3} + 1)}^{2}$ as $(\frac{2 x}{3} + 1) (\frac{2 x}{3} + 1)$ .

$x^{2} + (\frac{2 x}{3} + 1) (\frac{2 x}{3} + 1) = 5$

$y = \frac{2 x}{3} + 1$

Step 2.2.1.1.2

Expand $(\frac{2 x}{3} + 1) (\frac{2 x}{3} + 1)$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$x^{2} + \frac{2 x}{3} \cdot (\frac{2 x}{3} + 1) + 1 (\frac{2 x}{3} + 1) = 5$

$y = \frac{2 x}{3} + 1$

Step 2.2.1.1.2.2

Apply the distributive property.

$x^{2} + \frac{2 x}{3} \cdot \frac{2 x}{3} + \frac{2 x}{3} \cdot 1 + 1 (\frac{2 x}{3} + 1) = 5$

$y = \frac{2 x}{3} + 1$

Step 2.2.1.1.2.3

Apply the distributive property.

$x^{2} + \frac{2 x}{3} \cdot \frac{2 x}{3} + \frac{2 x}{3} \cdot 1 + 1 (\frac{2 x}{3}) + 1 \cdot 1 = 5$

$y = \frac{2 x}{3} + 1$

$x^{2} + \frac{2 x}{3} \cdot \frac{2 x}{3} + \frac{2 x}{3} \cdot 1 + 1 (\frac{2 x}{3}) + 1 \cdot 1 = 5$

$y = \frac{2 x}{3} + 1$

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 $\frac{2 x}{3} \cdot \frac{2 x}{3}$ .

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Step 2.2.1.1.3.1.1.1

Multiply $\frac{2 x}{3}$ by $\frac{2 x}{3}$ .

$x^{2} + \frac{2 x (2 x)}{3 \cdot 3} + \frac{2 x}{3} \cdot 1 + 1 (\frac{2 x}{3}) + 1 \cdot 1 = 5$

$y = \frac{2 x}{3} + 1$

Step 2.2.1.1.3.1.1.2

Multiply $2$ by $2$ .

$x^{2} + \frac{4 x \cdot x}{3 \cdot 3} + \frac{2 x}{3} \cdot 1 + 1 (\frac{2 x}{3}) + 1 \cdot 1 = 5$

$y = \frac{2 x}{3} + 1$

Step 2.2.1.1.3.1.1.3

Raise $x$ to the power of $1$ .

$x^{2} + \frac{4 (x \cdot x)}{3 \cdot 3} + \frac{2 x}{3} \cdot 1 + 1 (\frac{2 x}{3}) + 1 \cdot 1 = 5$

$y = \frac{2 x}{3} + 1$

Step 2.2.1.1.3.1.1.4

Raise $x$ to the power of $1$ .

$x^{2} + \frac{4 (x \cdot x)}{3 \cdot 3} + \frac{2 x}{3} \cdot 1 + 1 (\frac{2 x}{3}) + 1 \cdot 1 = 5$

$y = \frac{2 x}{3} + 1$

Step 2.2.1.1.3.1.1.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2} + \frac{4 x^{1 + 1}}{3 \cdot 3} + \frac{2 x}{3} \cdot 1 + 1 (\frac{2 x}{3}) + 1 \cdot 1 = 5$

$y = \frac{2 x}{3} + 1$

Step 2.2.1.1.3.1.1.6

Add $1$ and $1$ .

$x^{2} + \frac{4 x^{2}}{3 \cdot 3} + \frac{2 x}{3} \cdot 1 + 1 (\frac{2 x}{3}) + 1 \cdot 1 = 5$

$y = \frac{2 x}{3} + 1$

Step 2.2.1.1.3.1.1.7

Multiply $3$ by $3$ .

$x^{2} + \frac{4 x^{2}}{9} + \frac{2 x}{3} \cdot 1 + 1 (\frac{2 x}{3}) + 1 \cdot 1 = 5$

$y = \frac{2 x}{3} + 1$

$x^{2} + \frac{4 x^{2}}{9} + \frac{2 x}{3} \cdot 1 + 1 (\frac{2 x}{3}) + 1 \cdot 1 = 5$

$y = \frac{2 x}{3} + 1$

Step 2.2.1.1.3.1.2

Multiply $\frac{2 x}{3}$ by $1$ .

$x^{2} + \frac{4 x^{2}}{9} + \frac{2 x}{3} + 1 (\frac{2 x}{3}) + 1 \cdot 1 = 5$

$y = \frac{2 x}{3} + 1$

Step 2.2.1.1.3.1.3

Multiply $\frac{2 x}{3}$ by $1$ .

$x^{2} + \frac{4 x^{2}}{9} + \frac{2 x}{3} + \frac{2 x}{3} + 1 \cdot 1 = 5$

$y = \frac{2 x}{3} + 1$

Step 2.2.1.1.3.1.4

Multiply $1$ by $1$ .

$x^{2} + \frac{4 x^{2}}{9} + \frac{2 x}{3} + \frac{2 x}{3} + 1 = 5$

$y = \frac{2 x}{3} + 1$

$x^{2} + \frac{4 x^{2}}{9} + \frac{2 x}{3} + \frac{2 x}{3} + 1 = 5$

$y = \frac{2 x}{3} + 1$

Step 2.2.1.1.3.2

Add $\frac{2 x}{3}$ and $\frac{2 x}{3}$ .

$x^{2} + \frac{4 x^{2}}{9} + 2 (\frac{2 x}{3}) + 1 = 5$

$y = \frac{2 x}{3} + 1$

$x^{2} + \frac{4 x^{2}}{9} + 2 (\frac{2 x}{3}) + 1 = 5$

$y = \frac{2 x}{3} + 1$

Step 2.2.1.1.4

Multiply $2 \frac{2 x}{3}$ .

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Step 2.2.1.1.4.1

Combine $2$ and $\frac{2 x}{3}$ .

$x^{2} + \frac{4 x^{2}}{9} + \frac{2 (2 x)}{3} + 1 = 5$

$y = \frac{2 x}{3} + 1$

Step 2.2.1.1.4.2

Multiply $2$ by $2$ .

$x^{2} + \frac{4 x^{2}}{9} + \frac{4 x}{3} + 1 = 5$

$y = \frac{2 x}{3} + 1$

$x^{2} + \frac{4 x^{2}}{9} + \frac{4 x}{3} + 1 = 5$

$y = \frac{2 x}{3} + 1$

$x^{2} + \frac{4 x^{2}}{9} + \frac{4 x}{3} + 1 = 5$

$y = \frac{2 x}{3} + 1$

Step 2.2.1.2

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$x^{2} \cdot \frac{9}{9} + \frac{4 x^{2}}{9} + \frac{4 x}{3} + 1 = 5$

$y = \frac{2 x}{3} + 1$

Step 2.2.1.3

Combine $x^{2}$ and $\frac{9}{9}$ .

$\frac{x^{2} \cdot 9}{9} + \frac{4 x^{2}}{9} + \frac{4 x}{3} + 1 = 5$

$y = \frac{2 x}{3} + 1$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 9 + 4 x^{2}}{9} + \frac{4 x}{3} + 1 = 5$

$y = \frac{2 x}{3} + 1$

Step 2.2.1.5

Add $x^{2} \cdot 9$ and $4 x^{2}$ .

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Step 2.2.1.5.1

Reorder $x^{2}$ and $9$ .

$\frac{9 \cdot x^{2} + 4 x^{2}}{9} + \frac{4 x}{3} + 1 = 5$

$y = \frac{2 x}{3} + 1$

Step 2.2.1.5.2

Add $9 \cdot x^{2}$ and $4 x^{2}$ .

$\frac{13 \cdot x^{2}}{9} + \frac{4 x}{3} + 1 = 5$

$y = \frac{2 x}{3} + 1$

$\frac{13 x^{2}}{9} + \frac{4 x}{3} + 1 = 5$

$y = \frac{2 x}{3} + 1$

$\frac{13 x^{2}}{9} + \frac{4 x}{3} + 1 = 5$

$y = \frac{2 x}{3} + 1$

$\frac{13 x^{2}}{9} + \frac{4 x}{3} + 1 = 5$

$y = \frac{2 x}{3} + 1$

$\frac{13 x^{2}}{9} + \frac{4 x}{3} + 1 = 5$

$y = \frac{2 x}{3} + 1$

Step 3

Solve for $x$ in $\frac{13 x^{2}}{9} + \frac{4 x}{3} + 1 = 5$ .

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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 $5$ from both sides of the equation.

$\frac{13 x^{2}}{9} + \frac{4 x}{3} + 1 - 5 = 0$

$y = \frac{2 x}{3} + 1$

Step 3.1.2

Subtract $5$ from $1$ .

$\frac{13 x^{2}}{9} + \frac{4 x}{3} - 4 = 0$

$y = \frac{2 x}{3} + 1$

$\frac{13 x^{2}}{9} + \frac{4 x}{3} - 4 = 0$

$y = \frac{2 x}{3} + 1$

Step 3.2

Multiply through by the least common denominator $9$ , then simplify.

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Step 3.2.1

Apply the distributive property.

$9 (\frac{13 x^{2}}{9}) + 9 (\frac{4 x}{3}) + 9 \cdot - 4 = 0$

$y = \frac{2 x}{3} + 1$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Cancel the common factor of $9$ .

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Step 3.2.2.1.1

Cancel the common factor.

$9 (\frac{13 x^{2}}{9}) + 9 (\frac{4 x}{3}) + 9 \cdot - 4 = 0$

$y = \frac{2 x}{3} + 1$

Step 3.2.2.1.2

Rewrite the expression.

$13 x^{2} + 9 (\frac{4 x}{3}) + 9 \cdot - 4 = 0$

$y = \frac{2 x}{3} + 1$

$13 x^{2} + 9 (\frac{4 x}{3}) + 9 \cdot - 4 = 0$

$y = \frac{2 x}{3} + 1$

Step 3.2.2.2

Cancel the common factor of $3$ .

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Step 3.2.2.2.1

Factor $3$ out of $9$ .

$13 x^{2} + 3 (3) (\frac{4 x}{3}) + 9 \cdot - 4 = 0$

$y = \frac{2 x}{3} + 1$

Step 3.2.2.2.2

Cancel the common factor.

$13 x^{2} + 3 \cdot (3 (\frac{4 x}{3})) + 9 \cdot - 4 = 0$

$y = \frac{2 x}{3} + 1$

Step 3.2.2.2.3

Rewrite the expression.

$13 x^{2} + 3 (4 x) + 9 \cdot - 4 = 0$

$y = \frac{2 x}{3} + 1$

$13 x^{2} + 3 (4 x) + 9 \cdot - 4 = 0$

$y = \frac{2 x}{3} + 1$

Step 3.2.2.3

Multiply $4$ by $3$ .

$13 x^{2} + 12 x + 9 \cdot - 4 = 0$

$y = \frac{2 x}{3} + 1$

Step 3.2.2.4

Multiply $9$ by $- 4$ .

$13 x^{2} + 12 x - 36 = 0$

$y = \frac{2 x}{3} + 1$

$13 x^{2} + 12 x - 36 = 0$

$y = \frac{2 x}{3} + 1$

$13 x^{2} + 12 x - 36 = 0$

$y = \frac{2 x}{3} + 1$

Step 3.3

Use the quadratic formula to find the solutions.

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

$y = \frac{2 x}{3} + 1$

Step 3.4

Substitute the values $a = 13$ , $b = 12$ , and $c = - 36$ into the quadratic formula and solve for $x$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (13 \cdot - 36)}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.5

Simplify.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 13 \cdot - 36}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.5.1.2

Multiply $- 4 \cdot 13 \cdot - 36$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $13$ .

$x = \frac{- 12 \pm \sqrt{144 - 52 \cdot - 36}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.5.1.2.2

Multiply $- 52$ by $- 36$ .

$x = \frac{- 12 \pm \sqrt{144 + 1872}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

$x = \frac{- 12 \pm \sqrt{144 + 1872}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.5.1.3

Add $144$ and $1872$ .

$x = \frac{- 12 \pm \sqrt{2016}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.5.1.4

Rewrite $2016$ as $12^{2} \cdot 14$ .

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Step 3.5.1.4.1

Factor $144$ out of $2016$ .

$x = \frac{- 12 \pm \sqrt{144 (14)}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.5.1.4.2

Rewrite $144$ as $12^{2}$ .

$x = \frac{- 12 \pm \sqrt{12^{2} \cdot 14}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

$x = \frac{- 12 \pm \sqrt{12^{2} \cdot 14}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.5.1.5

Pull terms out from under the radical.

$x = \frac{- 12 \pm 12 \sqrt{14}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

$x = \frac{- 12 \pm 12 \sqrt{14}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.5.2

Multiply $2$ by $13$ .

$x = \frac{- 12 \pm 12 \sqrt{14}}{26}$

$y = \frac{2 x}{3} + 1$

Step 3.5.3

Simplify $\frac{- 12 \pm 12 \sqrt{14}}{26}$ .

$x = \frac{- 6 \pm 6 \sqrt{14}}{13}$

$y = \frac{2 x}{3} + 1$

$x = \frac{- 6 \pm 6 \sqrt{14}}{13}$

$y = \frac{2 x}{3} + 1$

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 $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 13 \cdot - 36}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.6.1.2

Multiply $- 4 \cdot 13 \cdot - 36$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $13$ .

$x = \frac{- 12 \pm \sqrt{144 - 52 \cdot - 36}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.6.1.2.2

Multiply $- 52$ by $- 36$ .

$x = \frac{- 12 \pm \sqrt{144 + 1872}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

$x = \frac{- 12 \pm \sqrt{144 + 1872}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.6.1.3

Add $144$ and $1872$ .

$x = \frac{- 12 \pm \sqrt{2016}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.6.1.4

Rewrite $2016$ as $12^{2} \cdot 14$ .

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Step 3.6.1.4.1

Factor $144$ out of $2016$ .

$x = \frac{- 12 \pm \sqrt{144 (14)}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.6.1.4.2

Rewrite $144$ as $12^{2}$ .

$x = \frac{- 12 \pm \sqrt{12^{2} \cdot 14}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

$x = \frac{- 12 \pm \sqrt{12^{2} \cdot 14}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.6.1.5

Pull terms out from under the radical.

$x = \frac{- 12 \pm 12 \sqrt{14}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

$x = \frac{- 12 \pm 12 \sqrt{14}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.6.2

Multiply $2$ by $13$ .

$x = \frac{- 12 \pm 12 \sqrt{14}}{26}$

$y = \frac{2 x}{3} + 1$

Step 3.6.3

Simplify $\frac{- 12 \pm 12 \sqrt{14}}{26}$ .

$x = \frac{- 6 \pm 6 \sqrt{14}}{13}$

$y = \frac{2 x}{3} + 1$

Step 3.6.4

Change the $\pm$ to $+$ .

$x = \frac{- 6 + 6 \sqrt{14}}{13}$

$y = \frac{2 x}{3} + 1$

Step 3.6.5

Rewrite $- 6$ as $- 1 (6)$ .

$x = \frac{- 1 \cdot 6 + 6 \sqrt{14}}{13}$

$y = \frac{2 x}{3} + 1$

Step 3.6.6

Factor $- 1$ out of $6 \sqrt{14}$ .

$x = \frac{- 1 \cdot 6 - (- 6 \sqrt{14})}{13}$

$y = \frac{2 x}{3} + 1$

Step 3.6.7

Factor $- 1$ out of $- 1 (6) - (- 6 \sqrt{14})$ .

$x = \frac{- 1 (6 - 6 \sqrt{14})}{13}$

$y = \frac{2 x}{3} + 1$

Step 3.6.8

Move the negative in front of the fraction.

$x = - \frac{6 - 6 \sqrt{14}}{13}$

$y = \frac{2 x}{3} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

$y = \frac{2 x}{3} + 1$

Step 3.7

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

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 13 \cdot - 36}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.7.1.2

Multiply $- 4 \cdot 13 \cdot - 36$ .

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Step 3.7.1.2.1

Multiply $- 4$ by $13$ .

$x = \frac{- 12 \pm \sqrt{144 - 52 \cdot - 36}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.7.1.2.2

Multiply $- 52$ by $- 36$ .

$x = \frac{- 12 \pm \sqrt{144 + 1872}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

$x = \frac{- 12 \pm \sqrt{144 + 1872}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.7.1.3

Add $144$ and $1872$ .

$x = \frac{- 12 \pm \sqrt{2016}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.7.1.4

Rewrite $2016$ as $12^{2} \cdot 14$ .

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Step 3.7.1.4.1

Factor $144$ out of $2016$ .

$x = \frac{- 12 \pm \sqrt{144 (14)}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.7.1.4.2

Rewrite $144$ as $12^{2}$ .

$x = \frac{- 12 \pm \sqrt{12^{2} \cdot 14}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

$x = \frac{- 12 \pm \sqrt{12^{2} \cdot 14}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.7.1.5

Pull terms out from under the radical.

$x = \frac{- 12 \pm 12 \sqrt{14}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

$x = \frac{- 12 \pm 12 \sqrt{14}}{2 \cdot 13}$

$y = \frac{2 x}{3} + 1$

Step 3.7.2

Multiply $2$ by $13$ .

$x = \frac{- 12 \pm 12 \sqrt{14}}{26}$

$y = \frac{2 x}{3} + 1$

Step 3.7.3

Simplify $\frac{- 12 \pm 12 \sqrt{14}}{26}$ .

$x = \frac{- 6 \pm 6 \sqrt{14}}{13}$

$y = \frac{2 x}{3} + 1$

Step 3.7.4

Change the $\pm$ to $-$ .

$x = \frac{- 6 - 6 \sqrt{14}}{13}$

$y = \frac{2 x}{3} + 1$

Step 3.7.5

Rewrite $- 6$ as $- 1 (6)$ .

$x = \frac{- 1 \cdot 6 - 6 \sqrt{14}}{13}$

$y = \frac{2 x}{3} + 1$

Step 3.7.6

Factor $- 1$ out of $- 6 \sqrt{14}$ .

$x = \frac{- 1 \cdot 6 - (6 \sqrt{14})}{13}$

$y = \frac{2 x}{3} + 1$

Step 3.7.7

Factor $- 1$ out of $- 1 (6) - (6 \sqrt{14})$ .

$x = \frac{- 1 (6 + 6 \sqrt{14})}{13}$

$y = \frac{2 x}{3} + 1$

Step 3.7.8

Move the negative in front of the fraction.

$x = - \frac{6 + 6 \sqrt{14}}{13}$

$y = \frac{2 x}{3} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

$y = \frac{2 x}{3} + 1$

Step 3.8

The final answer is the combination of both solutions.

$x = - \frac{6 - 6 \sqrt{14}}{13}, - \frac{6 + 6 \sqrt{14}}{13}$

$y = \frac{2 x}{3} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}, - \frac{6 + 6 \sqrt{14}}{13}$

$y = \frac{2 x}{3} + 1$

Step 4

Replace all occurrences of $x$ with $- \frac{6 - 6 \sqrt{14}}{13}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = \frac{2 x}{3} + 1$ with $- \frac{6 - 6 \sqrt{14}}{13}$ .

$y = \frac{2 (- \frac{6 - 6 \sqrt{14}}{13})}{3} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{2 (- \frac{6 - 6 \sqrt{14}}{13})}{3} + 1$ .

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Step 4.2.1.1

Simplify terms.

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Step 4.2.1.1.1

Simplify each term.

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Step 4.2.1.1.1.1

Simplify the numerator.

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Step 4.2.1.1.1.1.1

Multiply $- 1$ by $2$ .

$y = \frac{- 2 \frac{6 - 6 \sqrt{14}}{13}}{3} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 4.2.1.1.1.1.2

Combine $- 2$ and $\frac{6 - 6 \sqrt{14}}{13}$ .

$y = \frac{\frac{- 2 (6 - 6 \sqrt{14})}{13}}{3} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

$y = \frac{\frac{- 2 (6 - 6 \sqrt{14})}{13}}{3} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 4.2.1.1.1.2

Move the negative in front of the fraction.

$y = \frac{- \frac{2 (6 - 6 \sqrt{14})}{13}}{3} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 4.2.1.1.1.3

Simplify the numerator.

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Step 4.2.1.1.1.3.1

Apply the distributive property.

$y = \frac{- \frac{2 \cdot 6 + 2 (- 6 \sqrt{14})}{13}}{3} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 4.2.1.1.1.3.2

Multiply $2$ by $6$ .

$y = \frac{- \frac{12 + 2 (- 6 \sqrt{14})}{13}}{3} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 4.2.1.1.1.3.3

Multiply $- 6$ by $2$ .

$y = \frac{- \frac{12 - 12 \sqrt{14}}{13}}{3} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

$y = \frac{- \frac{12 - 12 \sqrt{14}}{13}}{3} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 4.2.1.1.1.4

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{12 - 12 \sqrt{14}}{13} \cdot \frac{1}{3} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 4.2.1.1.1.5

Multiply $- \frac{12 - 12 \sqrt{14}}{13} \cdot \frac{1}{3}$ .

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Step 4.2.1.1.1.5.1

Multiply $\frac{1}{3}$ by $\frac{12 - 12 \sqrt{14}}{13}$ .

$y = - \frac{12 - 12 \sqrt{14}}{3 \cdot 13} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 4.2.1.1.1.5.2

Multiply $3$ by $13$ .

$y = - \frac{12 - 12 \sqrt{14}}{39} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

$y = - \frac{12 - 12 \sqrt{14}}{39} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 4.2.1.1.1.6

Cancel the common factor of $12 - 12 \sqrt{14}$ and $39$ .

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Step 4.2.1.1.1.6.1

Factor $3$ out of $12$ .

$y = - \frac{3 (4) - 12 \sqrt{14}}{39} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 4.2.1.1.1.6.2

Factor $3$ out of $- 12 \sqrt{14}$ .

$y = - \frac{3 (4) + 3 (- 4 \sqrt{14})}{39} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 4.2.1.1.1.6.3

Factor $3$ out of $3 (4) + 3 (- 4 \sqrt{14})$ .

$y = - \frac{3 (4 - 4 \sqrt{14})}{39} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 4.2.1.1.1.6.4

Cancel the common factors.

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Step 4.2.1.1.1.6.4.1

Factor $3$ out of $39$ .

$y = - \frac{3 (4 - 4 \sqrt{14})}{3 \cdot 13} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 4.2.1.1.1.6.4.2

Cancel the common factor.

$y = - \frac{3 (4 - 4 \sqrt{14})}{3 \cdot 13} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 4.2.1.1.1.6.4.3

Rewrite the expression.

$y = - \frac{4 - 4 \sqrt{14}}{13} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

$y = - \frac{4 - 4 \sqrt{14}}{13} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

$y = - \frac{4 - 4 \sqrt{14}}{13} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

$y = - \frac{4 - 4 \sqrt{14}}{13} + 1$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 4.2.1.1.2

Simplify the expression.

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Step 4.2.1.1.2.1

Write $1$ as a fraction with a common denominator.

$y = - \frac{4 - 4 \sqrt{14}}{13} + \frac{13}{13}$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 4.2.1.1.2.2

Combine the numerators over the common denominator.

$y = \frac{- (4 - 4 \sqrt{14}) + 13}{13}$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

$y = \frac{- (4 - 4 \sqrt{14}) + 13}{13}$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

$y = \frac{- (4 - 4 \sqrt{14}) + 13}{13}$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 4.2.1.2

Simplify the numerator.

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Step 4.2.1.2.1

Apply the distributive property.

$y = \frac{- 1 \cdot 4 - (- 4 \sqrt{14}) + 13}{13}$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 4.2.1.2.2

Multiply $- 1$ by $4$ .

$y = \frac{- 4 - (- 4 \sqrt{14}) + 13}{13}$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 4.2.1.2.3

Multiply $- 4$ by $- 1$ .

$y = \frac{- 4 + 4 \sqrt{14} + 13}{13}$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 4.2.1.2.4

Add $- 4$ and $13$ .

$y = \frac{9 + 4 \sqrt{14}}{13}$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

$y = \frac{9 + 4 \sqrt{14}}{13}$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

$y = \frac{9 + 4 \sqrt{14}}{13}$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

$y = \frac{9 + 4 \sqrt{14}}{13}$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

$y = \frac{9 + 4 \sqrt{14}}{13}$

$x = - \frac{6 - 6 \sqrt{14}}{13}$

Step 5

Replace all occurrences of $x$ with $- \frac{6 + 6 \sqrt{14}}{13}$ in each equation.

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Step 5.1

Replace all occurrences of $x$ in $y = \frac{2 x}{3} + 1$ with $- \frac{6 + 6 \sqrt{14}}{13}$ .

$y = \frac{2 (- \frac{6 + 6 \sqrt{14}}{13})}{3} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{2 (- \frac{6 + 6 \sqrt{14}}{13})}{3} + 1$ .

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Step 5.2.1.1

Simplify terms.

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Step 5.2.1.1.1

Simplify each term.

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Step 5.2.1.1.1.1

Simplify the numerator.

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Step 5.2.1.1.1.1.1

Multiply $- 1$ by $2$ .

$y = \frac{- 2 \frac{6 + 6 \sqrt{14}}{13}}{3} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 5.2.1.1.1.1.2

Combine $- 2$ and $\frac{6 + 6 \sqrt{14}}{13}$ .

$y = \frac{\frac{- 2 (6 + 6 \sqrt{14})}{13}}{3} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

$y = \frac{\frac{- 2 (6 + 6 \sqrt{14})}{13}}{3} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 5.2.1.1.1.2

Move the negative in front of the fraction.

$y = \frac{- \frac{2 (6 + 6 \sqrt{14})}{13}}{3} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 5.2.1.1.1.3

Simplify the numerator.

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Step 5.2.1.1.1.3.1

Apply the distributive property.

$y = \frac{- \frac{2 \cdot 6 + 2 (6 \sqrt{14})}{13}}{3} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 5.2.1.1.1.3.2

Multiply $2$ by $6$ .

$y = \frac{- \frac{12 + 2 (6 \sqrt{14})}{13}}{3} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 5.2.1.1.1.3.3

Multiply $6$ by $2$ .

$y = \frac{- \frac{12 + 12 \sqrt{14}}{13}}{3} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

$y = \frac{- \frac{12 + 12 \sqrt{14}}{13}}{3} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 5.2.1.1.1.4

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{12 + 12 \sqrt{14}}{13} \cdot \frac{1}{3} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 5.2.1.1.1.5

Multiply $- \frac{12 + 12 \sqrt{14}}{13} \cdot \frac{1}{3}$ .

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Step 5.2.1.1.1.5.1

Multiply $\frac{1}{3}$ by $\frac{12 + 12 \sqrt{14}}{13}$ .

$y = - \frac{12 + 12 \sqrt{14}}{3 \cdot 13} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 5.2.1.1.1.5.2

Multiply $3$ by $13$ .

$y = - \frac{12 + 12 \sqrt{14}}{39} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

$y = - \frac{12 + 12 \sqrt{14}}{39} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 5.2.1.1.1.6

Cancel the common factor of $12 + 12 \sqrt{14}$ and $39$ .

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Step 5.2.1.1.1.6.1

Factor $3$ out of $12$ .

$y = - \frac{3 (4) + 12 \sqrt{14}}{39} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 5.2.1.1.1.6.2

Factor $3$ out of $12 \sqrt{14}$ .

$y = - \frac{3 (4) + 3 (4 \sqrt{14})}{39} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 5.2.1.1.1.6.3

Factor $3$ out of $3 (4) + 3 (4 \sqrt{14})$ .

$y = - \frac{3 (4 + 4 \sqrt{14})}{39} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 5.2.1.1.1.6.4

Cancel the common factors.

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Step 5.2.1.1.1.6.4.1

Factor $3$ out of $39$ .

$y = - \frac{3 (4 + 4 \sqrt{14})}{3 \cdot 13} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 5.2.1.1.1.6.4.2

Cancel the common factor.

$y = - \frac{3 (4 + 4 \sqrt{14})}{3 \cdot 13} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 5.2.1.1.1.6.4.3

Rewrite the expression.

$y = - \frac{4 + 4 \sqrt{14}}{13} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

$y = - \frac{4 + 4 \sqrt{14}}{13} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

$y = - \frac{4 + 4 \sqrt{14}}{13} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

$y = - \frac{4 + 4 \sqrt{14}}{13} + 1$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 5.2.1.1.2

Simplify the expression.

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Step 5.2.1.1.2.1

Write $1$ as a fraction with a common denominator.

$y = - \frac{4 + 4 \sqrt{14}}{13} + \frac{13}{13}$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 5.2.1.1.2.2

Combine the numerators over the common denominator.

$y = \frac{- (4 + 4 \sqrt{14}) + 13}{13}$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

$y = \frac{- (4 + 4 \sqrt{14}) + 13}{13}$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

$y = \frac{- (4 + 4 \sqrt{14}) + 13}{13}$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 5.2.1.2

Simplify the numerator.

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Step 5.2.1.2.1

Apply the distributive property.

$y = \frac{- 1 \cdot 4 - (4 \sqrt{14}) + 13}{13}$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 5.2.1.2.2

Multiply $- 1$ by $4$ .

$y = \frac{- 4 - (4 \sqrt{14}) + 13}{13}$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 5.2.1.2.3

Multiply $4$ by $- 1$ .

$y = \frac{- 4 - 4 \sqrt{14} + 13}{13}$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 5.2.1.2.4

Add $- 4$ and $13$ .

$y = \frac{9 - 4 \sqrt{14}}{13}$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

$y = \frac{9 - 4 \sqrt{14}}{13}$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

$y = \frac{9 - 4 \sqrt{14}}{13}$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

$y = \frac{9 - 4 \sqrt{14}}{13}$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

$y = \frac{9 - 4 \sqrt{14}}{13}$

$x = - \frac{6 + 6 \sqrt{14}}{13}$

Step 6

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

$(- \frac{6 - 6 \sqrt{14}}{13}, \frac{9 + 4 \sqrt{14}}{13})$

$(- \frac{6 + 6 \sqrt{14}}{13}, \frac{9 - 4 \sqrt{14}}{13})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(- \frac{6 - 6 \sqrt{14}}{13}, \frac{9 + 4 \sqrt{14}}{13}), (- \frac{6 + 6 \sqrt{14}}{13}, \frac{9 - 4 \sqrt{14}}{13})$

Equation Form:

$x = - \frac{6 - 6 \sqrt{14}}{13}, y = \frac{9 + 4 \sqrt{14}}{13}$

$x = - \frac{6 + 6 \sqrt{14}}{13}, y = \frac{9 - 4 \sqrt{14}}{13}$

Step 8