Algebra Examples

$x = 2 y + 5$ $y = (2 x - 3) (x + 9)$

Step 1

Replace all occurrences of $x$ with $2 y + 5$ in each equation.

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Step 1.1

Replace all occurrences of $x$ in $y = (2 x - 3) (x + 9)$ with $2 y + 5$ .

$y = (2 (2 y + 5) - 3) ((2 y + 5) + 9)$

$x = 2 y + 5$

Step 1.2

Simplify the right side.

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Step 1.2.1

Simplify $(2 (2 y + 5) - 3) ((2 y + 5) + 9)$ .

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Step 1.2.1.1

Simplify terms.

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Step 1.2.1.1.1

Simplify each term.

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Step 1.2.1.1.1.1

Apply the distributive property.

$y = (2 (2 y) + 2 \cdot 5 - 3) ((2 y + 5) + 9)$

$x = 2 y + 5$

Step 1.2.1.1.1.2

Multiply $2$ by $2$ .

$y = (4 y + 2 \cdot 5 - 3) ((2 y + 5) + 9)$

$x = 2 y + 5$

Step 1.2.1.1.1.3

Multiply $2$ by $5$ .

$y = (4 y + 10 - 3) ((2 y + 5) + 9)$

$x = 2 y + 5$

$y = (4 y + 10 - 3) ((2 y + 5) + 9)$

$x = 2 y + 5$

Step 1.2.1.1.2

Simplify by adding and subtracting.

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Step 1.2.1.1.2.1

Subtract $3$ from $10$ .

$y = (4 y + 7) ((2 y + 5) + 9)$

$x = 2 y + 5$

Step 1.2.1.1.2.2

Add $5$ and $9$ .

$y = (4 y + 7) (2 y + 14)$

$x = 2 y + 5$

$y = (4 y + 7) (2 y + 14)$

$x = 2 y + 5$

$y = (4 y + 7) (2 y + 14)$

$x = 2 y + 5$

Step 1.2.1.2

Expand $(4 y + 7) (2 y + 14)$ using the FOIL Method.

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Step 1.2.1.2.1

Apply the distributive property.

$y = 4 y (2 y + 14) + 7 (2 y + 14)$

$x = 2 y + 5$

Step 1.2.1.2.2

Apply the distributive property.

$y = 4 y (2 y) + 4 y \cdot 14 + 7 (2 y + 14)$

$x = 2 y + 5$

Step 1.2.1.2.3

Apply the distributive property.

$y = 4 y (2 y) + 4 y \cdot 14 + 7 (2 y) + 7 \cdot 14$

$x = 2 y + 5$

$y = 4 y (2 y) + 4 y \cdot 14 + 7 (2 y) + 7 \cdot 14$

$x = 2 y + 5$

Step 1.2.1.3

Simplify and combine like terms.

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Step 1.2.1.3.1

Simplify each term.

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Step 1.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$y = 4 \cdot (2 y \cdot y) + 4 y \cdot 14 + 7 (2 y) + 7 \cdot 14$

$x = 2 y + 5$

Step 1.2.1.3.1.2

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

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Step 1.2.1.3.1.2.1

Move $y$ .

$y = 4 \cdot (2 (y \cdot y)) + 4 y \cdot 14 + 7 (2 y) + 7 \cdot 14$

$x = 2 y + 5$

Step 1.2.1.3.1.2.2

Multiply $y$ by $y$ .

$y = 4 \cdot (2 y^{2}) + 4 y \cdot 14 + 7 (2 y) + 7 \cdot 14$

$x = 2 y + 5$

$y = 4 \cdot (2 y^{2}) + 4 y \cdot 14 + 7 (2 y) + 7 \cdot 14$

$x = 2 y + 5$

Step 1.2.1.3.1.3

Multiply $4$ by $2$ .

$y = 8 y^{2} + 4 y \cdot 14 + 7 (2 y) + 7 \cdot 14$

$x = 2 y + 5$

Step 1.2.1.3.1.4

Multiply $14$ by $4$ .

$y = 8 y^{2} + 56 y + 7 (2 y) + 7 \cdot 14$

$x = 2 y + 5$

Step 1.2.1.3.1.5

Multiply $2$ by $7$ .

$y = 8 y^{2} + 56 y + 14 y + 7 \cdot 14$

$x = 2 y + 5$

Step 1.2.1.3.1.6

Multiply $7$ by $14$ .

$y = 8 y^{2} + 56 y + 14 y + 98$

$x = 2 y + 5$

$y = 8 y^{2} + 56 y + 14 y + 98$

$x = 2 y + 5$

Step 1.2.1.3.2

Add $56 y$ and $14 y$ .

$y = 8 y^{2} + 70 y + 98$

$x = 2 y + 5$

$y = 8 y^{2} + 70 y + 98$

$x = 2 y + 5$

$y = 8 y^{2} + 70 y + 98$

$x = 2 y + 5$

$y = 8 y^{2} + 70 y + 98$

$x = 2 y + 5$

$y = 8 y^{2} + 70 y + 98$

$x = 2 y + 5$

Step 2

Solve for $y$ in $y = 8 y^{2} + 70 y + 98$ .

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Step 2.1

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$8 y^{2} + 70 y + 98 = y$

$x = 2 y + 5$

Step 2.2

Move all terms containing $y$ to the left side of the equation.

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Step 2.2.1

Subtract $y$ from both sides of the equation.

$8 y^{2} + 70 y + 98 - y = 0$

$x = 2 y + 5$

Step 2.2.2

Subtract $y$ from $70 y$ .

$8 y^{2} + 69 y + 98 = 0$

$x = 2 y + 5$

$8 y^{2} + 69 y + 98 = 0$

$x = 2 y + 5$

Step 2.3

Use the quadratic formula to find the solutions.

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

$x = 2 y + 5$

Step 2.4

Substitute the values $a = 8$ , $b = 69$ , and $c = 98$ into the quadratic formula and solve for $y$ .

$\frac{- 69 \pm \sqrt{69^{2} - 4 \cdot (8 \cdot 98)}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $69$ to the power of $2$ .

$y = \frac{- 69 \pm \sqrt{4761 - 4 \cdot 8 \cdot 98}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.5.1.2

Multiply $- 4 \cdot 8 \cdot 98$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $8$ .

$y = \frac{- 69 \pm \sqrt{4761 - 32 \cdot 98}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.5.1.2.2

Multiply $- 32$ by $98$ .

$y = \frac{- 69 \pm \sqrt{4761 - 3136}}{2 \cdot 8}$

$x = 2 y + 5$

$y = \frac{- 69 \pm \sqrt{4761 - 3136}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.5.1.3

Subtract $3136$ from $4761$ .

$y = \frac{- 69 \pm \sqrt{1625}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.5.1.4

Rewrite $1625$ as $5^{2} \cdot 65$ .

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Step 2.5.1.4.1

Factor $25$ out of $1625$ .

$y = \frac{- 69 \pm \sqrt{25 (65)}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.5.1.4.2

Rewrite $25$ as $5^{2}$ .

$y = \frac{- 69 \pm \sqrt{5^{2} \cdot 65}}{2 \cdot 8}$

$x = 2 y + 5$

$y = \frac{- 69 \pm \sqrt{5^{2} \cdot 65}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.5.1.5

Pull terms out from under the radical.

$y = \frac{- 69 \pm 5 \sqrt{65}}{2 \cdot 8}$

$x = 2 y + 5$

$y = \frac{- 69 \pm 5 \sqrt{65}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.5.2

Multiply $2$ by $8$ .

$y = \frac{- 69 \pm 5 \sqrt{65}}{16}$

$x = 2 y + 5$

$y = \frac{- 69 \pm 5 \sqrt{65}}{16}$

$x = 2 y + 5$

Step 2.6

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

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise $69$ to the power of $2$ .

$y = \frac{- 69 \pm \sqrt{4761 - 4 \cdot 8 \cdot 98}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.6.1.2

Multiply $- 4 \cdot 8 \cdot 98$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $8$ .

$y = \frac{- 69 \pm \sqrt{4761 - 32 \cdot 98}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.6.1.2.2

Multiply $- 32$ by $98$ .

$y = \frac{- 69 \pm \sqrt{4761 - 3136}}{2 \cdot 8}$

$x = 2 y + 5$

$y = \frac{- 69 \pm \sqrt{4761 - 3136}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.6.1.3

Subtract $3136$ from $4761$ .

$y = \frac{- 69 \pm \sqrt{1625}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.6.1.4

Rewrite $1625$ as $5^{2} \cdot 65$ .

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Step 2.6.1.4.1

Factor $25$ out of $1625$ .

$y = \frac{- 69 \pm \sqrt{25 (65)}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.6.1.4.2

Rewrite $25$ as $5^{2}$ .

$y = \frac{- 69 \pm \sqrt{5^{2} \cdot 65}}{2 \cdot 8}$

$x = 2 y + 5$

$y = \frac{- 69 \pm \sqrt{5^{2} \cdot 65}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.6.1.5

Pull terms out from under the radical.

$y = \frac{- 69 \pm 5 \sqrt{65}}{2 \cdot 8}$

$x = 2 y + 5$

$y = \frac{- 69 \pm 5 \sqrt{65}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.6.2

Multiply $2$ by $8$ .

$y = \frac{- 69 \pm 5 \sqrt{65}}{16}$

$x = 2 y + 5$

Step 2.6.3

Change the $\pm$ to $+$ .

$y = \frac{- 69 + 5 \sqrt{65}}{16}$

$x = 2 y + 5$

Step 2.6.4

Rewrite $- 69$ as $- 1 (69)$ .

$y = \frac{- 1 \cdot 69 + 5 \sqrt{65}}{16}$

$x = 2 y + 5$

Step 2.6.5

Factor $- 1$ out of $5 \sqrt{65}$ .

$y = \frac{- 1 \cdot 69 - (- 5 \sqrt{65})}{16}$

$x = 2 y + 5$

Step 2.6.6

Factor $- 1$ out of $- 1 (69) - (- 5 \sqrt{65})$ .

$y = \frac{- 1 (69 - 5 \sqrt{65})}{16}$

$x = 2 y + 5$

Step 2.6.7

Move the negative in front of the fraction.

$y = - \frac{69 - 5 \sqrt{65}}{16}$

$x = 2 y + 5$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

$x = 2 y + 5$

Step 2.7

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

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Raise $69$ to the power of $2$ .

$y = \frac{- 69 \pm \sqrt{4761 - 4 \cdot 8 \cdot 98}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.7.1.2

Multiply $- 4 \cdot 8 \cdot 98$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $8$ .

$y = \frac{- 69 \pm \sqrt{4761 - 32 \cdot 98}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.7.1.2.2

Multiply $- 32$ by $98$ .

$y = \frac{- 69 \pm \sqrt{4761 - 3136}}{2 \cdot 8}$

$x = 2 y + 5$

$y = \frac{- 69 \pm \sqrt{4761 - 3136}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.7.1.3

Subtract $3136$ from $4761$ .

$y = \frac{- 69 \pm \sqrt{1625}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.7.1.4

Rewrite $1625$ as $5^{2} \cdot 65$ .

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Step 2.7.1.4.1

Factor $25$ out of $1625$ .

$y = \frac{- 69 \pm \sqrt{25 (65)}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.7.1.4.2

Rewrite $25$ as $5^{2}$ .

$y = \frac{- 69 \pm \sqrt{5^{2} \cdot 65}}{2 \cdot 8}$

$x = 2 y + 5$

$y = \frac{- 69 \pm \sqrt{5^{2} \cdot 65}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.7.1.5

Pull terms out from under the radical.

$y = \frac{- 69 \pm 5 \sqrt{65}}{2 \cdot 8}$

$x = 2 y + 5$

$y = \frac{- 69 \pm 5 \sqrt{65}}{2 \cdot 8}$

$x = 2 y + 5$

Step 2.7.2

Multiply $2$ by $8$ .

$y = \frac{- 69 \pm 5 \sqrt{65}}{16}$

$x = 2 y + 5$

Step 2.7.3

Change the $\pm$ to $-$ .

$y = \frac{- 69 - 5 \sqrt{65}}{16}$

$x = 2 y + 5$

Step 2.7.4

Rewrite $- 69$ as $- 1 (69)$ .

$y = \frac{- 1 \cdot 69 - 5 \sqrt{65}}{16}$

$x = 2 y + 5$

Step 2.7.5

Factor $- 1$ out of $- 5 \sqrt{65}$ .

$y = \frac{- 1 \cdot 69 - (5 \sqrt{65})}{16}$

$x = 2 y + 5$

Step 2.7.6

Factor $- 1$ out of $- 1 (69) - (5 \sqrt{65})$ .

$y = \frac{- 1 (69 + 5 \sqrt{65})}{16}$

$x = 2 y + 5$

Step 2.7.7

Move the negative in front of the fraction.

$y = - \frac{69 + 5 \sqrt{65}}{16}$

$x = 2 y + 5$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

$x = 2 y + 5$

Step 2.8

The final answer is the combination of both solutions.

$y = - \frac{69 - 5 \sqrt{65}}{16}, - \frac{69 + 5 \sqrt{65}}{16}$

$x = 2 y + 5$

$y = - \frac{69 - 5 \sqrt{65}}{16}, - \frac{69 + 5 \sqrt{65}}{16}$

$x = 2 y + 5$

Step 3

Replace all occurrences of $y$ with $- \frac{69 - 5 \sqrt{65}}{16}$ in each equation.

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Step 3.1

Replace all occurrences of $y$ in $x = 2 y + 5$ with $- \frac{69 - 5 \sqrt{65}}{16}$ .

$x = 2 (- \frac{69 - 5 \sqrt{65}}{16}) + 5$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $2 (- \frac{69 - 5 \sqrt{65}}{16}) + 5$ .

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Step 3.2.1.1

Simplify each term.

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Step 3.2.1.1.1

Cancel the common factor of $2$ .

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Step 3.2.1.1.1.1

Move the leading negative in $- \frac{69 - 5 \sqrt{65}}{16}$ into the numerator.

$x = 2 (\frac{- (69 - 5 \sqrt{65})}{16}) + 5$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

Step 3.2.1.1.1.2

Factor $2$ out of $16$ .

$x = 2 (\frac{- (69 - 5 \sqrt{65})}{2 (8)}) + 5$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

Step 3.2.1.1.1.3

Cancel the common factor.

$x = 2 (\frac{- (69 - 5 \sqrt{65})}{2 \cdot 8}) + 5$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

Step 3.2.1.1.1.4

Rewrite the expression.

$x = \frac{- (69 - 5 \sqrt{65})}{8} + 5$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

$x = \frac{- (69 - 5 \sqrt{65})}{8} + 5$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

Step 3.2.1.1.2

Move the negative in front of the fraction.

$x = - \frac{69 - 5 \sqrt{65}}{8} + 5$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

$x = - \frac{69 - 5 \sqrt{65}}{8} + 5$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

Step 3.2.1.2

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

$x = - \frac{69 - 5 \sqrt{65}}{8} + 5 \cdot \frac{8}{8}$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

Step 3.2.1.3

Combine fractions.

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Step 3.2.1.3.1

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

$x = - \frac{69 - 5 \sqrt{65}}{8} + \frac{5 \cdot 8}{8}$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

Step 3.2.1.3.2

Combine the numerators over the common denominator.

$x = \frac{- (69 - 5 \sqrt{65}) + 5 \cdot 8}{8}$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

$x = \frac{- (69 - 5 \sqrt{65}) + 5 \cdot 8}{8}$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

Step 3.2.1.4

Simplify the numerator.

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Step 3.2.1.4.1

Apply the distributive property.

$x = \frac{- 1 \cdot 69 - (- 5 \sqrt{65}) + 5 \cdot 8}{8}$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

Step 3.2.1.4.2

Multiply $- 1$ by $69$ .

$x = \frac{- 69 - (- 5 \sqrt{65}) + 5 \cdot 8}{8}$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

Step 3.2.1.4.3

Multiply $- 5$ by $- 1$ .

$x = \frac{- 69 + 5 \sqrt{65} + 5 \cdot 8}{8}$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

Step 3.2.1.4.4

Multiply $5$ by $8$ .

$x = \frac{- 69 + 5 \sqrt{65} + 40}{8}$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

Step 3.2.1.4.5

Add $- 69$ and $40$ .

$x = \frac{- 29 + 5 \sqrt{65}}{8}$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

$x = \frac{- 29 + 5 \sqrt{65}}{8}$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

Step 3.2.1.5

Simplify with factoring out.

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Step 3.2.1.5.1

Rewrite $- 29$ as $- 1 (29)$ .

$x = \frac{- 1 \cdot 29 + 5 \sqrt{65}}{8}$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

Step 3.2.1.5.2

Factor $- 1$ out of $5 \sqrt{65}$ .

$x = \frac{- 1 \cdot 29 - (- 5 \sqrt{65})}{8}$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

Step 3.2.1.5.3

Factor $- 1$ out of $- 1 (29) - (- 5 \sqrt{65})$ .

$x = \frac{- 1 (29 - 5 \sqrt{65})}{8}$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

Step 3.2.1.5.4

Move the negative in front of the fraction.

$x = - \frac{29 - 5 \sqrt{65}}{8}$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

$x = - \frac{29 - 5 \sqrt{65}}{8}$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

$x = - \frac{29 - 5 \sqrt{65}}{8}$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

$x = - \frac{29 - 5 \sqrt{65}}{8}$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

$x = - \frac{29 - 5 \sqrt{65}}{8}$

$y = - \frac{69 - 5 \sqrt{65}}{16}$

Step 4

Replace all occurrences of $y$ with $- \frac{69 + 5 \sqrt{65}}{16}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 2 y + 5$ with $- \frac{69 + 5 \sqrt{65}}{16}$ .

$x = 2 (- \frac{69 + 5 \sqrt{65}}{16}) + 5$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $2 (- \frac{69 + 5 \sqrt{65}}{16}) + 5$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Cancel the common factor of $2$ .

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Step 4.2.1.1.1.1

Move the leading negative in $- \frac{69 + 5 \sqrt{65}}{16}$ into the numerator.

$x = 2 (\frac{- (69 + 5 \sqrt{65})}{16}) + 5$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

Step 4.2.1.1.1.2

Factor $2$ out of $16$ .

$x = 2 (\frac{- (69 + 5 \sqrt{65})}{2 (8)}) + 5$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

Step 4.2.1.1.1.3

Cancel the common factor.

$x = 2 (\frac{- (69 + 5 \sqrt{65})}{2 \cdot 8}) + 5$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

Step 4.2.1.1.1.4

Rewrite the expression.

$x = \frac{- (69 + 5 \sqrt{65})}{8} + 5$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

$x = \frac{- (69 + 5 \sqrt{65})}{8} + 5$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

Step 4.2.1.1.2

Move the negative in front of the fraction.

$x = - \frac{69 + 5 \sqrt{65}}{8} + 5$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

$x = - \frac{69 + 5 \sqrt{65}}{8} + 5$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

Step 4.2.1.2

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

$x = - \frac{69 + 5 \sqrt{65}}{8} + 5 \cdot \frac{8}{8}$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

Step 4.2.1.3

Combine fractions.

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Step 4.2.1.3.1

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

$x = - \frac{69 + 5 \sqrt{65}}{8} + \frac{5 \cdot 8}{8}$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

Step 4.2.1.3.2

Combine the numerators over the common denominator.

$x = \frac{- (69 + 5 \sqrt{65}) + 5 \cdot 8}{8}$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

$x = \frac{- (69 + 5 \sqrt{65}) + 5 \cdot 8}{8}$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

Step 4.2.1.4

Simplify the numerator.

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Step 4.2.1.4.1

Apply the distributive property.

$x = \frac{- 1 \cdot 69 - (5 \sqrt{65}) + 5 \cdot 8}{8}$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

Step 4.2.1.4.2

Multiply $- 1$ by $69$ .

$x = \frac{- 69 - (5 \sqrt{65}) + 5 \cdot 8}{8}$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

Step 4.2.1.4.3

Multiply $5$ by $- 1$ .

$x = \frac{- 69 - 5 \sqrt{65} + 5 \cdot 8}{8}$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

Step 4.2.1.4.4

Multiply $5$ by $8$ .

$x = \frac{- 69 - 5 \sqrt{65} + 40}{8}$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

Step 4.2.1.4.5

Add $- 69$ and $40$ .

$x = \frac{- 29 - 5 \sqrt{65}}{8}$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

$x = \frac{- 29 - 5 \sqrt{65}}{8}$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

Step 4.2.1.5

Simplify with factoring out.

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Step 4.2.1.5.1

Rewrite $- 29$ as $- 1 (29)$ .

$x = \frac{- 1 \cdot 29 - 5 \sqrt{65}}{8}$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

Step 4.2.1.5.2

Factor $- 1$ out of $- 5 \sqrt{65}$ .

$x = \frac{- 1 \cdot 29 - (5 \sqrt{65})}{8}$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

Step 4.2.1.5.3

Factor $- 1$ out of $- 1 (29) - (5 \sqrt{65})$ .

$x = \frac{- 1 (29 + 5 \sqrt{65})}{8}$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

Step 4.2.1.5.4

Move the negative in front of the fraction.

$x = - \frac{29 + 5 \sqrt{65}}{8}$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

$x = - \frac{29 + 5 \sqrt{65}}{8}$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

$x = - \frac{29 + 5 \sqrt{65}}{8}$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

$x = - \frac{29 + 5 \sqrt{65}}{8}$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

$x = - \frac{29 + 5 \sqrt{65}}{8}$

$y = - \frac{69 + 5 \sqrt{65}}{16}$

Step 5

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

$(- \frac{29 - 5 \sqrt{65}}{8}, - \frac{69 - 5 \sqrt{65}}{16})$

$(- \frac{29 + 5 \sqrt{65}}{8}, - \frac{69 + 5 \sqrt{65}}{16})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- \frac{29 - 5 \sqrt{65}}{8}, - \frac{69 - 5 \sqrt{65}}{16}), (- \frac{29 + 5 \sqrt{65}}{8}, - \frac{69 + 5 \sqrt{65}}{16})$

Equation Form:

$x = - \frac{29 - 5 \sqrt{65}}{8}, y = - \frac{69 - 5 \sqrt{65}}{16}$

$x = - \frac{29 + 5 \sqrt{65}}{8}, y = - \frac{69 + 5 \sqrt{65}}{16}$

Step 7