Precalculus Examples

$x^{2} + y^{2} = 16$ , $4 x + 7 y = 13$

Step 1

Solve for $x$ in $4 x + 7 y = 13$ .

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Step 1.1

Subtract $7 y$ from both sides of the equation.

$4 x = 13 - 7 y$

$x^{2} + y^{2} = 16$

Step 1.2

Divide each term in $4 x = 13 - 7 y$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 x = 13 - 7 y$ by $4$ .

$\frac{4 x}{4} = \frac{13}{4} + \frac{- 7 y}{4}$

$x^{2} + y^{2} = 16$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{13}{4} + \frac{- 7 y}{4}$

$x^{2} + y^{2} = 16$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{13}{4} + \frac{- 7 y}{4}$

$x^{2} + y^{2} = 16$

$x = \frac{13}{4} + \frac{- 7 y}{4}$

$x^{2} + y^{2} = 16$

$x = \frac{13}{4} + \frac{- 7 y}{4}$

$x^{2} + y^{2} = 16$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$x = \frac{13}{4} - \frac{7 y}{4}$

$x^{2} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$x^{2} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$x^{2} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$x^{2} + y^{2} = 16$

Step 2

Replace all occurrences of $x$ with $\frac{13}{4} - \frac{7 y}{4}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x^{2} + y^{2} = 16$ with $\frac{13}{4} - \frac{7 y}{4}$ .

${(\frac{13}{4} - \frac{7 y}{4})}^{2} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(\frac{13}{4} - \frac{7 y}{4})}^{2} + y^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(\frac{13}{4} - \frac{7 y}{4})}^{2}$ as $(\frac{13}{4} - \frac{7 y}{4}) (\frac{13}{4} - \frac{7 y}{4})$ .

$(\frac{13}{4} - \frac{7 y}{4}) (\frac{13}{4} - \frac{7 y}{4}) + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.2

Expand $(\frac{13}{4} - \frac{7 y}{4}) (\frac{13}{4} - \frac{7 y}{4})$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$\frac{13}{4} \cdot (\frac{13}{4} - \frac{7 y}{4}) - \frac{7 y}{4} \cdot (\frac{13}{4} - \frac{7 y}{4}) + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.2.2

Apply the distributive property.

$\frac{13}{4} \cdot \frac{13}{4} + \frac{13}{4} \cdot (- \frac{7 y}{4}) - \frac{7 y}{4} \cdot (\frac{13}{4} - \frac{7 y}{4}) + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.2.3

Apply the distributive property.

$\frac{13}{4} \cdot \frac{13}{4} + \frac{13}{4} \cdot (- \frac{7 y}{4}) - \frac{7 y}{4} \cdot \frac{13}{4} - \frac{7 y}{4} \cdot (- \frac{7 y}{4}) + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$\frac{13}{4} \cdot \frac{13}{4} + \frac{13}{4} \cdot (- \frac{7 y}{4}) - \frac{7 y}{4} \cdot \frac{13}{4} - \frac{7 y}{4} \cdot (- \frac{7 y}{4}) + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

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 $\frac{13}{4} \cdot \frac{13}{4}$ .

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Step 2.2.1.1.3.1.1.1

Multiply $\frac{13}{4}$ by $\frac{13}{4}$ .

$\frac{13 \cdot 13}{4 \cdot 4} + \frac{13}{4} \cdot (- \frac{7 y}{4}) - \frac{7 y}{4} \cdot \frac{13}{4} - \frac{7 y}{4} \cdot (- \frac{7 y}{4}) + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.3.1.1.2

Multiply $13$ by $13$ .

$\frac{169}{4 \cdot 4} + \frac{13}{4} \cdot (- \frac{7 y}{4}) - \frac{7 y}{4} \cdot \frac{13}{4} - \frac{7 y}{4} \cdot (- \frac{7 y}{4}) + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.3.1.1.3

Multiply $4$ by $4$ .

$\frac{169}{16} + \frac{13}{4} \cdot (- \frac{7 y}{4}) - \frac{7 y}{4} \cdot \frac{13}{4} - \frac{7 y}{4} \cdot (- \frac{7 y}{4}) + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$\frac{169}{16} + \frac{13}{4} \cdot (- \frac{7 y}{4}) - \frac{7 y}{4} \cdot \frac{13}{4} - \frac{7 y}{4} \cdot (- \frac{7 y}{4}) + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.3.1.2

Multiply $\frac{13}{4} (- \frac{7 y}{4})$ .

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Step 2.2.1.1.3.1.2.1

Multiply $\frac{13}{4}$ by $\frac{7 y}{4}$ .

$\frac{169}{16} - \frac{13 (7 y)}{4 \cdot 4} - \frac{7 y}{4} \cdot \frac{13}{4} - \frac{7 y}{4} \cdot (- \frac{7 y}{4}) + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.3.1.2.2

Multiply $7$ by $13$ .

$\frac{169}{16} - \frac{91 y}{4 \cdot 4} - \frac{7 y}{4} \cdot \frac{13}{4} - \frac{7 y}{4} \cdot (- \frac{7 y}{4}) + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.3.1.2.3

Multiply $4$ by $4$ .

$\frac{169}{16} - \frac{91 y}{16} - \frac{7 y}{4} \cdot \frac{13}{4} - \frac{7 y}{4} \cdot (- \frac{7 y}{4}) + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$\frac{169}{16} - \frac{91 y}{16} - \frac{7 y}{4} \cdot \frac{13}{4} - \frac{7 y}{4} \cdot (- \frac{7 y}{4}) + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.3.1.3

Multiply $- \frac{7 y}{4} \cdot \frac{13}{4}$ .

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Step 2.2.1.1.3.1.3.1

Multiply $\frac{13}{4}$ by $\frac{7 y}{4}$ .

$\frac{169}{16} - \frac{91 y}{16} - \frac{13 (7 y)}{4 \cdot 4} - \frac{7 y}{4} \cdot (- \frac{7 y}{4}) + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.3.1.3.2

Multiply $7$ by $13$ .

$\frac{169}{16} - \frac{91 y}{16} - \frac{91 y}{4 \cdot 4} - \frac{7 y}{4} \cdot (- \frac{7 y}{4}) + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.3.1.3.3

Multiply $4$ by $4$ .

$\frac{169}{16} - \frac{91 y}{16} - \frac{91 y}{16} - \frac{7 y}{4} \cdot (- \frac{7 y}{4}) + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$\frac{169}{16} - \frac{91 y}{16} - \frac{91 y}{16} - \frac{7 y}{4} \cdot (- \frac{7 y}{4}) + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.3.1.4

Multiply $- \frac{7 y}{4} (- \frac{7 y}{4})$ .

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Step 2.2.1.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{169}{16} - \frac{91 y}{16} - \frac{91 y}{16} + 1 (\frac{7 y}{4}) \cdot \frac{7 y}{4} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.3.1.4.2

Multiply $\frac{7 y}{4}$ by $1$ .

$\frac{169}{16} - \frac{91 y}{16} - \frac{91 y}{16} + \frac{7 y}{4} \cdot \frac{7 y}{4} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.3.1.4.3

Multiply $\frac{7 y}{4}$ by $\frac{7 y}{4}$ .

$\frac{169}{16} - \frac{91 y}{16} - \frac{91 y}{16} + \frac{7 y (7 y)}{4 \cdot 4} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.3.1.4.4

Multiply $7$ by $7$ .

$\frac{169}{16} - \frac{91 y}{16} - \frac{91 y}{16} + \frac{49 y \cdot y}{4 \cdot 4} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.3.1.4.5

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

$\frac{169}{16} - \frac{91 y}{16} - \frac{91 y}{16} + \frac{49 (y \cdot y)}{4 \cdot 4} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.3.1.4.6

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

$\frac{169}{16} - \frac{91 y}{16} - \frac{91 y}{16} + \frac{49 (y \cdot y)}{4 \cdot 4} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.3.1.4.7

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

$\frac{169}{16} - \frac{91 y}{16} - \frac{91 y}{16} + \frac{49 y^{1 + 1}}{4 \cdot 4} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.3.1.4.8

Add $1$ and $1$ .

$\frac{169}{16} - \frac{91 y}{16} - \frac{91 y}{16} + \frac{49 y^{2}}{4 \cdot 4} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.3.1.4.9

Multiply $4$ by $4$ .

$\frac{169}{16} - \frac{91 y}{16} - \frac{91 y}{16} + \frac{49 y^{2}}{16} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$\frac{169}{16} - \frac{91 y}{16} - \frac{91 y}{16} + \frac{49 y^{2}}{16} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$\frac{169}{16} - \frac{91 y}{16} - \frac{91 y}{16} + \frac{49 y^{2}}{16} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.3.2

Subtract $\frac{91 y}{16}$ from $- \frac{91 y}{16}$ .

$\frac{169}{16} - 2 \frac{91 y}{16} + \frac{49 y^{2}}{16} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$\frac{169}{16} - 2 \frac{91 y}{16} + \frac{49 y^{2}}{16} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.4

Simplify each term.

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Step 2.2.1.1.4.1

Cancel the common factor of $2$ .

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Step 2.2.1.1.4.1.1

Factor $2$ out of $- 2$ .

$\frac{169}{16} + 2 (- 1) (\frac{91 y}{16}) + \frac{49 y^{2}}{16} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.4.1.2

Factor $2$ out of $16$ .

$\frac{169}{16} + 2 \cdot (- 1 \frac{91 y}{2 \cdot 8}) + \frac{49 y^{2}}{16} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.4.1.3

Cancel the common factor.

$\frac{169}{16} + 2 \cdot (- 1 \frac{91 y}{2 \cdot 8}) + \frac{49 y^{2}}{16} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.4.1.4

Rewrite the expression.

$\frac{169}{16} - 1 \frac{91 y}{8} + \frac{49 y^{2}}{16} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$\frac{169}{16} - 1 \frac{91 y}{8} + \frac{49 y^{2}}{16} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.1.4.2

Rewrite $- 1 \frac{91 y}{8}$ as $- \frac{91 y}{8}$ .

$\frac{169}{16} - \frac{91 y}{8} + \frac{49 y^{2}}{16} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$\frac{169}{16} - \frac{91 y}{8} + \frac{49 y^{2}}{16} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$\frac{169}{16} - \frac{91 y}{8} + \frac{49 y^{2}}{16} + y^{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.2

To write $y^{2}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{169}{16} - \frac{91 y}{8} + \frac{49 y^{2}}{16} + y^{2} \cdot \frac{16}{16} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

Combine $y^{2}$ and $\frac{16}{16}$ .

$\frac{169}{16} - \frac{91 y}{8} + \frac{49 y^{2}}{16} + \frac{y^{2} \cdot 16}{16} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$\frac{169}{16} - \frac{91 y}{8} + \frac{49 y^{2} + y^{2} \cdot 16}{16} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.3.3

Combine the numerators over the common denominator.

$\frac{169 + 49 y^{2} + y^{2} \cdot 16}{16} + \frac{- 91 y}{8} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$\frac{169 + 49 y^{2} + y^{2} \cdot 16}{16} + \frac{- 91 y}{8} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.4

Move $16$ to the left of $y^{2}$ .

$\frac{169 + 49 y^{2} + 16 y^{2}}{16} + \frac{- 91 y}{8} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.5

Add $49 y^{2}$ and $16 y^{2}$ .

$\frac{169 + 65 y^{2}}{16} + \frac{- 91 y}{8} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.6

Simplify each term.

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Step 2.2.1.6.1

Factor $13$ out of $169 + 65 y^{2}$ .

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Step 2.2.1.6.1.1

Factor $13$ out of $169$ .

$\frac{13 (13) + 65 y^{2}}{16} + \frac{- 91 y}{8} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.6.1.2

Factor $13$ out of $65 y^{2}$ .

$\frac{13 (13) + 13 (5 y^{2})}{16} + \frac{- 91 y}{8} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.6.1.3

Factor $13$ out of $13 (13) + 13 (5 y^{2})$ .

$\frac{13 (13 + 5 y^{2})}{16} + \frac{- 91 y}{8} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$\frac{13 (13 + 5 y^{2})}{16} + \frac{- 91 y}{8} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.6.2

Move the negative in front of the fraction.

$\frac{13 (13 + 5 y^{2})}{16} - \frac{91 y}{8} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$\frac{13 (13 + 5 y^{2})}{16} - \frac{91 y}{8} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.7

To write $- \frac{91 y}{8}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{13 (13 + 5 y^{2})}{16} - \frac{91 y}{8} \cdot \frac{2}{2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.8

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.8.1

Multiply $\frac{91 y}{8}$ by $\frac{2}{2}$ .

$\frac{13 (13 + 5 y^{2})}{16} - \frac{91 y \cdot 2}{8 \cdot 2} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.8.2

Multiply $8$ by $2$ .

$\frac{13 (13 + 5 y^{2})}{16} - \frac{91 y \cdot 2}{16} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$\frac{13 (13 + 5 y^{2})}{16} - \frac{91 y \cdot 2}{16} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.9

Combine the numerators over the common denominator.

$\frac{13 (13 + 5 y^{2}) - 91 y \cdot 2}{16} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.10

Simplify the numerator.

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Step 2.2.1.10.1

Factor $13$ out of $13 (13 + 5 y^{2}) - 91 y \cdot 2$ .

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Step 2.2.1.10.1.1

Factor $13$ out of $- 91 y \cdot 2$ .

$\frac{13 (13 + 5 y^{2}) + 13 (- 7 y \cdot 2)}{16} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.10.1.2

Factor $13$ out of $13 (13 + 5 y^{2}) + 13 (- 7 y \cdot 2)$ .

$\frac{13 (13 + 5 y^{2} - 7 y \cdot 2)}{16} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$\frac{13 (13 + 5 y^{2} - 7 y \cdot 2)}{16} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.10.2

Multiply $2$ by $- 7$ .

$\frac{13 (13 + 5 y^{2} - 14 y)}{16} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 2.2.1.10.3

Reorder terms.

$\frac{13 (5 y^{2} - 14 y + 13)}{16} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$\frac{13 (5 y^{2} - 14 y + 13)}{16} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$\frac{13 (5 y^{2} - 14 y + 13)}{16} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$\frac{13 (5 y^{2} - 14 y + 13)}{16} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$\frac{13 (5 y^{2} - 14 y + 13)}{16} = 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3

Solve for $y$ in $\frac{13 (5 y^{2} - 14 y + 13)}{16} = 16$ .

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Step 3.1

Multiply both sides by $16$ .

$\frac{13 (5 y^{2} - 14 y + 13)}{16} \cdot 16 = 16 \cdot 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $\frac{13 (5 y^{2} - 14 y + 13)}{16} \cdot 16$ .

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Step 3.2.1.1.1

Cancel the common factor of $16$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{13 (5 y^{2} - 14 y + 13)}{16} \cdot 16 = 16 \cdot 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.2.1.1.1.2

Rewrite the expression.

$13 (5 y^{2} - 14 y + 13) = 16 \cdot 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$13 (5 y^{2} - 14 y + 13) = 16 \cdot 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.2.1.1.2

Apply the distributive property.

$13 (5 y^{2}) + 13 (- 14 y) + 13 \cdot 13 = 16 \cdot 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.2.1.1.3

Simplify.

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Step 3.2.1.1.3.1

Multiply $5$ by $13$ .

$65 y^{2} + 13 (- 14 y) + 13 \cdot 13 = 16 \cdot 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.2.1.1.3.2

Multiply $- 14$ by $13$ .

$65 y^{2} - 182 y + 13 \cdot 13 = 16 \cdot 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.2.1.1.3.3

Multiply $13$ by $13$ .

$65 y^{2} - 182 y + 169 = 16 \cdot 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$65 y^{2} - 182 y + 169 = 16 \cdot 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$65 y^{2} - 182 y + 169 = 16 \cdot 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

$65 y^{2} - 182 y + 169 = 16 \cdot 16$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $16$ by $16$ .

$65 y^{2} - 182 y + 169 = 256$

$x = \frac{13}{4} - \frac{7 y}{4}$

$65 y^{2} - 182 y + 169 = 256$

$x = \frac{13}{4} - \frac{7 y}{4}$

$65 y^{2} - 182 y + 169 = 256$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Move all terms to the left side of the equation and simplify.

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Step 3.3.1.1

Subtract $256$ from both sides of the equation.

$65 y^{2} - 182 y + 169 - 256 = 0$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.1.2

Subtract $256$ from $169$ .

$65 y^{2} - 182 y - 87 = 0$

$x = \frac{13}{4} - \frac{7 y}{4}$

$65 y^{2} - 182 y - 87 = 0$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.2

Use the quadratic formula to find the solutions.

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

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.3

Substitute the values $a = 65$ , $b = - 182$ , and $c = - 87$ into the quadratic formula and solve for $y$ .

$\frac{182 \pm \sqrt{{(- 182)}^{2} - 4 \cdot (65 \cdot - 87)}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.4

Simplify.

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Step 3.3.4.1

Simplify the numerator.

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Step 3.3.4.1.1

Raise $- 182$ to the power of $2$ .

$y = \frac{182 \pm \sqrt{33124 - 4 \cdot 65 \cdot - 87}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.4.1.2

Multiply $- 4 \cdot 65 \cdot - 87$ .

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Step 3.3.4.1.2.1

Multiply $- 4$ by $65$ .

$y = \frac{182 \pm \sqrt{33124 - 260 \cdot - 87}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.4.1.2.2

Multiply $- 260$ by $- 87$ .

$y = \frac{182 \pm \sqrt{33124 + 22620}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

$y = \frac{182 \pm \sqrt{33124 + 22620}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.4.1.3

Add $33124$ and $22620$ .

$y = \frac{182 \pm \sqrt{55744}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.4.1.4

Rewrite $55744$ as $8^{2} \cdot 871$ .

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Step 3.3.4.1.4.1

Factor $64$ out of $55744$ .

$y = \frac{182 \pm \sqrt{64 (871)}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.4.1.4.2

Rewrite $64$ as $8^{2}$ .

$y = \frac{182 \pm \sqrt{8^{2} \cdot 871}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

$y = \frac{182 \pm \sqrt{8^{2} \cdot 871}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.4.1.5

Pull terms out from under the radical.

$y = \frac{182 \pm 8 \sqrt{871}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

$y = \frac{182 \pm 8 \sqrt{871}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.4.2

Multiply $2$ by $65$ .

$y = \frac{182 \pm 8 \sqrt{871}}{130}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.4.3

Simplify $\frac{182 \pm 8 \sqrt{871}}{130}$ .

$y = \frac{91 \pm 4 \sqrt{871}}{65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

$y = \frac{91 \pm 4 \sqrt{871}}{65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.5

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

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Step 3.3.5.1

Simplify the numerator.

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Step 3.3.5.1.1

Raise $- 182$ to the power of $2$ .

$y = \frac{182 \pm \sqrt{33124 - 4 \cdot 65 \cdot - 87}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.5.1.2

Multiply $- 4 \cdot 65 \cdot - 87$ .

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Step 3.3.5.1.2.1

Multiply $- 4$ by $65$ .

$y = \frac{182 \pm \sqrt{33124 - 260 \cdot - 87}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.5.1.2.2

Multiply $- 260$ by $- 87$ .

$y = \frac{182 \pm \sqrt{33124 + 22620}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

$y = \frac{182 \pm \sqrt{33124 + 22620}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.5.1.3

Add $33124$ and $22620$ .

$y = \frac{182 \pm \sqrt{55744}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.5.1.4

Rewrite $55744$ as $8^{2} \cdot 871$ .

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Step 3.3.5.1.4.1

Factor $64$ out of $55744$ .

$y = \frac{182 \pm \sqrt{64 (871)}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.5.1.4.2

Rewrite $64$ as $8^{2}$ .

$y = \frac{182 \pm \sqrt{8^{2} \cdot 871}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

$y = \frac{182 \pm \sqrt{8^{2} \cdot 871}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.5.1.5

Pull terms out from under the radical.

$y = \frac{182 \pm 8 \sqrt{871}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

$y = \frac{182 \pm 8 \sqrt{871}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.5.2

Multiply $2$ by $65$ .

$y = \frac{182 \pm 8 \sqrt{871}}{130}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.5.3

Simplify $\frac{182 \pm 8 \sqrt{871}}{130}$ .

$y = \frac{91 \pm 4 \sqrt{871}}{65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.5.4

Change the $\pm$ to $+$ .

$y = \frac{91 + 4 \sqrt{871}}{65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.6

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

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Step 3.3.6.1

Simplify the numerator.

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Step 3.3.6.1.1

Raise $- 182$ to the power of $2$ .

$y = \frac{182 \pm \sqrt{33124 - 4 \cdot 65 \cdot - 87}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.6.1.2

Multiply $- 4 \cdot 65 \cdot - 87$ .

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Step 3.3.6.1.2.1

Multiply $- 4$ by $65$ .

$y = \frac{182 \pm \sqrt{33124 - 260 \cdot - 87}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.6.1.2.2

Multiply $- 260$ by $- 87$ .

$y = \frac{182 \pm \sqrt{33124 + 22620}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

$y = \frac{182 \pm \sqrt{33124 + 22620}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.6.1.3

Add $33124$ and $22620$ .

$y = \frac{182 \pm \sqrt{55744}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.6.1.4

Rewrite $55744$ as $8^{2} \cdot 871$ .

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Step 3.3.6.1.4.1

Factor $64$ out of $55744$ .

$y = \frac{182 \pm \sqrt{64 (871)}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.6.1.4.2

Rewrite $64$ as $8^{2}$ .

$y = \frac{182 \pm \sqrt{8^{2} \cdot 871}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

$y = \frac{182 \pm \sqrt{8^{2} \cdot 871}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.6.1.5

Pull terms out from under the radical.

$y = \frac{182 \pm 8 \sqrt{871}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

$y = \frac{182 \pm 8 \sqrt{871}}{2 \cdot 65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.6.2

Multiply $2$ by $65$ .

$y = \frac{182 \pm 8 \sqrt{871}}{130}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.6.3

Simplify $\frac{182 \pm 8 \sqrt{871}}{130}$ .

$y = \frac{91 \pm 4 \sqrt{871}}{65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.6.4

Change the $\pm$ to $-$ .

$y = \frac{91 - 4 \sqrt{871}}{65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 3.3.7

The final answer is the combination of both solutions.

$y = \frac{91 + 4 \sqrt{871}}{65}, \frac{91 - 4 \sqrt{871}}{65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

$y = \frac{91 + 4 \sqrt{871}}{65}, \frac{91 - 4 \sqrt{871}}{65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

$y = \frac{91 + 4 \sqrt{871}}{65}, \frac{91 - 4 \sqrt{871}}{65}$

$x = \frac{13}{4} - \frac{7 y}{4}$

Step 4

Replace all occurrences of $y$ with $\frac{91 + 4 \sqrt{871}}{65}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{13}{4} - \frac{7 y}{4}$ with $\frac{91 + 4 \sqrt{871}}{65}$ .

$x = \frac{13}{4} - \frac{7 (\frac{91 + 4 \sqrt{871}}{65})}{4}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{13}{4} - \frac{7 (\frac{91 + 4 \sqrt{871}}{65})}{4}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{13 - 7 \frac{91 + 4 \sqrt{871}}{65}}{4}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

Step 4.2.1.2

Simplify each term.

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Step 4.2.1.2.1

Combine $- 7$ and $\frac{91 + 4 \sqrt{871}}{65}$ .

$x = \frac{13 + \frac{- 7 (91 + 4 \sqrt{871})}{65}}{4}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

Step 4.2.1.2.2

Move the negative in front of the fraction.

$x = \frac{13 - \frac{7 (91 + 4 \sqrt{871})}{65}}{4}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

$x = \frac{13 - \frac{7 (91 + 4 \sqrt{871})}{65}}{4}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

Step 4.2.1.3

To write $13$ as a fraction with a common denominator, multiply by $\frac{65}{65}$ .

$x = \frac{13 \cdot \frac{65}{65} - \frac{7 (91 + 4 \sqrt{871})}{65}}{4}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

Step 4.2.1.4

Combine fractions.

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Step 4.2.1.4.1

Combine $13$ and $\frac{65}{65}$ .

$x = \frac{\frac{13 \cdot 65}{65} - \frac{7 (91 + 4 \sqrt{871})}{65}}{4}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

Step 4.2.1.4.2

Combine the numerators over the common denominator.

$x = \frac{\frac{13 \cdot 65 - 7 (91 + 4 \sqrt{871})}{65}}{4}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

$x = \frac{\frac{13 \cdot 65 - 7 (91 + 4 \sqrt{871})}{65}}{4}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $13$ by $65$ .

$x = \frac{\frac{845 - 7 (91 + 4 \sqrt{871})}{65}}{4}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

Step 4.2.1.5.2

Apply the distributive property.

$x = \frac{\frac{845 - 7 \cdot 91 - 7 (4 \sqrt{871})}{65}}{4}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

Step 4.2.1.5.3

Multiply $- 7$ by $91$ .

$x = \frac{\frac{845 - 637 - 7 (4 \sqrt{871})}{65}}{4}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

Step 4.2.1.5.4

Multiply $4$ by $- 7$ .

$x = \frac{\frac{845 - 637 - 28 \sqrt{871}}{65}}{4}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

Step 4.2.1.5.5

Subtract $637$ from $845$ .

$x = \frac{\frac{208 - 28 \sqrt{871}}{65}}{4}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

$x = \frac{\frac{208 - 28 \sqrt{871}}{65}}{4}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

Step 4.2.1.6

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{208 - 28 \sqrt{871}}{65} \cdot \frac{1}{4}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

Step 4.2.1.7

Multiply $\frac{208 - 28 \sqrt{871}}{65} \cdot \frac{1}{4}$ .

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Step 4.2.1.7.1

Multiply $\frac{208 - 28 \sqrt{871}}{65}$ by $\frac{1}{4}$ .

$x = \frac{208 - 28 \sqrt{871}}{65 \cdot 4}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

Step 4.2.1.7.2

Multiply $65$ by $4$ .

$x = \frac{208 - 28 \sqrt{871}}{260}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

$x = \frac{208 - 28 \sqrt{871}}{260}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

Step 4.2.1.8

Cancel the common factor of $208 - 28 \sqrt{871}$ and $260$ .

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Step 4.2.1.8.1

Factor $4$ out of $208$ .

$x = \frac{4 (52) - 28 \sqrt{871}}{260}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

Step 4.2.1.8.2

Factor $4$ out of $- 28 \sqrt{871}$ .

$x = \frac{4 (52) + 4 (- 7 \sqrt{871})}{260}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

Step 4.2.1.8.3

Factor $4$ out of $4 (52) + 4 (- 7 \sqrt{871})$ .

$x = \frac{4 (52 - 7 \sqrt{871})}{260}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

Step 4.2.1.8.4

Cancel the common factors.

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Step 4.2.1.8.4.1

Factor $4$ out of $260$ .

$x = \frac{4 (52 - 7 \sqrt{871})}{4 \cdot 65}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

Step 4.2.1.8.4.2

Cancel the common factor.

$x = \frac{4 (52 - 7 \sqrt{871})}{4 \cdot 65}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

Step 4.2.1.8.4.3

Rewrite the expression.

$x = \frac{52 - 7 \sqrt{871}}{65}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

$x = \frac{52 - 7 \sqrt{871}}{65}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

$x = \frac{52 - 7 \sqrt{871}}{65}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

$x = \frac{52 - 7 \sqrt{871}}{65}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

$x = \frac{52 - 7 \sqrt{871}}{65}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

$x = \frac{52 - 7 \sqrt{871}}{65}$

$y = \frac{91 + 4 \sqrt{871}}{65}$

Step 5

Replace all occurrences of $y$ with $\frac{91 - 4 \sqrt{871}}{65}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = \frac{13}{4} - \frac{7 y}{4}$ with $\frac{91 - 4 \sqrt{871}}{65}$ .

$x = \frac{13}{4} - \frac{7 (\frac{91 - 4 \sqrt{871}}{65})}{4}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{13}{4} - \frac{7 (\frac{91 - 4 \sqrt{871}}{65})}{4}$ .

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Step 5.2.1.1

Combine the numerators over the common denominator.

$x = \frac{13 - 7 \frac{91 - 4 \sqrt{871}}{65}}{4}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

Step 5.2.1.2

Simplify each term.

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Step 5.2.1.2.1

Combine $- 7$ and $\frac{91 - 4 \sqrt{871}}{65}$ .

$x = \frac{13 + \frac{- 7 (91 - 4 \sqrt{871})}{65}}{4}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

Step 5.2.1.2.2

Move the negative in front of the fraction.

$x = \frac{13 - \frac{7 (91 - 4 \sqrt{871})}{65}}{4}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

$x = \frac{13 - \frac{7 (91 - 4 \sqrt{871})}{65}}{4}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

Step 5.2.1.3

To write $13$ as a fraction with a common denominator, multiply by $\frac{65}{65}$ .

$x = \frac{13 \cdot \frac{65}{65} - \frac{7 (91 - 4 \sqrt{871})}{65}}{4}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

Step 5.2.1.4

Combine fractions.

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Step 5.2.1.4.1

Combine $13$ and $\frac{65}{65}$ .

$x = \frac{\frac{13 \cdot 65}{65} - \frac{7 (91 - 4 \sqrt{871})}{65}}{4}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

Step 5.2.1.4.2

Combine the numerators over the common denominator.

$x = \frac{\frac{13 \cdot 65 - 7 (91 - 4 \sqrt{871})}{65}}{4}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

$x = \frac{\frac{13 \cdot 65 - 7 (91 - 4 \sqrt{871})}{65}}{4}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

Step 5.2.1.5

Simplify the numerator.

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Step 5.2.1.5.1

Multiply $13$ by $65$ .

$x = \frac{\frac{845 - 7 (91 - 4 \sqrt{871})}{65}}{4}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

Step 5.2.1.5.2

Apply the distributive property.

$x = \frac{\frac{845 - 7 \cdot 91 - 7 (- 4 \sqrt{871})}{65}}{4}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

Step 5.2.1.5.3

Multiply $- 7$ by $91$ .

$x = \frac{\frac{845 - 637 - 7 (- 4 \sqrt{871})}{65}}{4}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

Step 5.2.1.5.4

Multiply $- 4$ by $- 7$ .

$x = \frac{\frac{845 - 637 + 28 \sqrt{871}}{65}}{4}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

Step 5.2.1.5.5

Subtract $637$ from $845$ .

$x = \frac{\frac{208 + 28 \sqrt{871}}{65}}{4}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

$x = \frac{\frac{208 + 28 \sqrt{871}}{65}}{4}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

Step 5.2.1.6

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{208 + 28 \sqrt{871}}{65} \cdot \frac{1}{4}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

Step 5.2.1.7

Multiply $\frac{208 + 28 \sqrt{871}}{65} \cdot \frac{1}{4}$ .

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Step 5.2.1.7.1

Multiply $\frac{208 + 28 \sqrt{871}}{65}$ by $\frac{1}{4}$ .

$x = \frac{208 + 28 \sqrt{871}}{65 \cdot 4}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

Step 5.2.1.7.2

Multiply $65$ by $4$ .

$x = \frac{208 + 28 \sqrt{871}}{260}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

$x = \frac{208 + 28 \sqrt{871}}{260}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

Step 5.2.1.8

Cancel the common factor of $208 + 28 \sqrt{871}$ and $260$ .

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Step 5.2.1.8.1

Factor $4$ out of $208$ .

$x = \frac{4 (52) + 28 \sqrt{871}}{260}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

Step 5.2.1.8.2

Factor $4$ out of $28 \sqrt{871}$ .

$x = \frac{4 (52) + 4 (7 \sqrt{871})}{260}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

Step 5.2.1.8.3

Factor $4$ out of $4 (52) + 4 (7 \sqrt{871})$ .

$x = \frac{4 (52 + 7 \sqrt{871})}{260}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

Step 5.2.1.8.4

Cancel the common factors.

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Step 5.2.1.8.4.1

Factor $4$ out of $260$ .

$x = \frac{4 (52 + 7 \sqrt{871})}{4 \cdot 65}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

Step 5.2.1.8.4.2

Cancel the common factor.

$x = \frac{4 (52 + 7 \sqrt{871})}{4 \cdot 65}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

Step 5.2.1.8.4.3

Rewrite the expression.

$x = \frac{52 + 7 \sqrt{871}}{65}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

$x = \frac{52 + 7 \sqrt{871}}{65}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

$x = \frac{52 + 7 \sqrt{871}}{65}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

$x = \frac{52 + 7 \sqrt{871}}{65}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

$x = \frac{52 + 7 \sqrt{871}}{65}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

$x = \frac{52 + 7 \sqrt{871}}{65}$

$y = \frac{91 - 4 \sqrt{871}}{65}$

Step 6

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

$(\frac{52 - 7 \sqrt{871}}{65}, \frac{91 + 4 \sqrt{871}}{65})$

$(\frac{52 + 7 \sqrt{871}}{65}, \frac{91 - 4 \sqrt{871}}{65})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(\frac{52 - 7 \sqrt{871}}{65}, \frac{91 + 4 \sqrt{871}}{65}), (\frac{52 + 7 \sqrt{871}}{65}, \frac{91 - 4 \sqrt{871}}{65})$

Equation Form:

$x = \frac{52 - 7 \sqrt{871}}{65}, y = \frac{91 + 4 \sqrt{871}}{65}$

$x = \frac{52 + 7 \sqrt{871}}{65}, y = \frac{91 - 4 \sqrt{871}}{65}$

Step 8