Algebra Examples

$x^{2} + y^{2} = 9$ $9 x + 2 y = 16$

Step 1

Solve for $x$ in $9 x + 2 y = 16$ .

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Step 1.1

Subtract $2 y$ from both sides of the equation.

$9 x = 16 - 2 y$

$x^{2} + y^{2} = 9$

Step 1.2

Divide each term in $9 x = 16 - 2 y$ by $9$ and simplify.

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Step 1.2.1

Divide each term in $9 x = 16 - 2 y$ by $9$ .

$\frac{9 x}{9} = \frac{16}{9} + \frac{- 2 y}{9}$

$x^{2} + y^{2} = 9$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $9$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{16}{9} + \frac{- 2 y}{9}$

$x^{2} + y^{2} = 9$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{16}{9} + \frac{- 2 y}{9}$

$x^{2} + y^{2} = 9$

$x = \frac{16}{9} + \frac{- 2 y}{9}$

$x^{2} + y^{2} = 9$

$x = \frac{16}{9} + \frac{- 2 y}{9}$

$x^{2} + y^{2} = 9$

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.

$x = \frac{16}{9} - \frac{2 y}{9}$

$x^{2} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

$x^{2} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

$x^{2} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

$x^{2} + y^{2} = 9$

Step 2

Replace all occurrences of $x$ with $\frac{16}{9} - \frac{2 y}{9}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x^{2} + y^{2} = 9$ with $\frac{16}{9} - \frac{2 y}{9}$ .

${(\frac{16}{9} - \frac{2 y}{9})}^{2} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(\frac{16}{9} - \frac{2 y}{9})}^{2} + y^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(\frac{16}{9} - \frac{2 y}{9})}^{2}$ as $(\frac{16}{9} - \frac{2 y}{9}) (\frac{16}{9} - \frac{2 y}{9})$ .

$(\frac{16}{9} - \frac{2 y}{9}) (\frac{16}{9} - \frac{2 y}{9}) + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.2

Expand $(\frac{16}{9} - \frac{2 y}{9}) (\frac{16}{9} - \frac{2 y}{9})$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$\frac{16}{9} \cdot (\frac{16}{9} - \frac{2 y}{9}) - \frac{2 y}{9} \cdot (\frac{16}{9} - \frac{2 y}{9}) + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.2.2

Apply the distributive property.

$\frac{16}{9} \cdot \frac{16}{9} + \frac{16}{9} \cdot (- \frac{2 y}{9}) - \frac{2 y}{9} \cdot (\frac{16}{9} - \frac{2 y}{9}) + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.2.3

Apply the distributive property.

$\frac{16}{9} \cdot \frac{16}{9} + \frac{16}{9} \cdot (- \frac{2 y}{9}) - \frac{2 y}{9} \cdot \frac{16}{9} - \frac{2 y}{9} \cdot (- \frac{2 y}{9}) + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

$\frac{16}{9} \cdot \frac{16}{9} + \frac{16}{9} \cdot (- \frac{2 y}{9}) - \frac{2 y}{9} \cdot \frac{16}{9} - \frac{2 y}{9} \cdot (- \frac{2 y}{9}) + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

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 $\frac{16}{9} \cdot \frac{16}{9}$ .

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Step 2.2.1.1.3.1.1.1

Multiply $\frac{16}{9}$ by $\frac{16}{9}$ .

$\frac{16 \cdot 16}{9 \cdot 9} + \frac{16}{9} \cdot (- \frac{2 y}{9}) - \frac{2 y}{9} \cdot \frac{16}{9} - \frac{2 y}{9} \cdot (- \frac{2 y}{9}) + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.3.1.1.2

Multiply $16$ by $16$ .

$\frac{256}{9 \cdot 9} + \frac{16}{9} \cdot (- \frac{2 y}{9}) - \frac{2 y}{9} \cdot \frac{16}{9} - \frac{2 y}{9} \cdot (- \frac{2 y}{9}) + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.3.1.1.3

Multiply $9$ by $9$ .

$\frac{256}{81} + \frac{16}{9} \cdot (- \frac{2 y}{9}) - \frac{2 y}{9} \cdot \frac{16}{9} - \frac{2 y}{9} \cdot (- \frac{2 y}{9}) + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

$\frac{256}{81} + \frac{16}{9} \cdot (- \frac{2 y}{9}) - \frac{2 y}{9} \cdot \frac{16}{9} - \frac{2 y}{9} \cdot (- \frac{2 y}{9}) + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.3.1.2

Multiply $\frac{16}{9} (- \frac{2 y}{9})$ .

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Step 2.2.1.1.3.1.2.1

Multiply $\frac{16}{9}$ by $\frac{2 y}{9}$ .

$\frac{256}{81} - \frac{16 (2 y)}{9 \cdot 9} - \frac{2 y}{9} \cdot \frac{16}{9} - \frac{2 y}{9} \cdot (- \frac{2 y}{9}) + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.3.1.2.2

Multiply $2$ by $16$ .

$\frac{256}{81} - \frac{32 y}{9 \cdot 9} - \frac{2 y}{9} \cdot \frac{16}{9} - \frac{2 y}{9} \cdot (- \frac{2 y}{9}) + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.3.1.2.3

Multiply $9$ by $9$ .

$\frac{256}{81} - \frac{32 y}{81} - \frac{2 y}{9} \cdot \frac{16}{9} - \frac{2 y}{9} \cdot (- \frac{2 y}{9}) + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

$\frac{256}{81} - \frac{32 y}{81} - \frac{2 y}{9} \cdot \frac{16}{9} - \frac{2 y}{9} \cdot (- \frac{2 y}{9}) + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.3.1.3

Multiply $- \frac{2 y}{9} \cdot \frac{16}{9}$ .

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Step 2.2.1.1.3.1.3.1

Multiply $\frac{16}{9}$ by $\frac{2 y}{9}$ .

$\frac{256}{81} - \frac{32 y}{81} - \frac{16 (2 y)}{9 \cdot 9} - \frac{2 y}{9} \cdot (- \frac{2 y}{9}) + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.3.1.3.2

Multiply $2$ by $16$ .

$\frac{256}{81} - \frac{32 y}{81} - \frac{32 y}{9 \cdot 9} - \frac{2 y}{9} \cdot (- \frac{2 y}{9}) + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.3.1.3.3

Multiply $9$ by $9$ .

$\frac{256}{81} - \frac{32 y}{81} - \frac{32 y}{81} - \frac{2 y}{9} \cdot (- \frac{2 y}{9}) + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

$\frac{256}{81} - \frac{32 y}{81} - \frac{32 y}{81} - \frac{2 y}{9} \cdot (- \frac{2 y}{9}) + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.3.1.4

Multiply $- \frac{2 y}{9} (- \frac{2 y}{9})$ .

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Step 2.2.1.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{256}{81} - \frac{32 y}{81} - \frac{32 y}{81} + 1 (\frac{2 y}{9}) \cdot \frac{2 y}{9} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.3.1.4.2

Multiply $\frac{2 y}{9}$ by $1$ .

$\frac{256}{81} - \frac{32 y}{81} - \frac{32 y}{81} + \frac{2 y}{9} \cdot \frac{2 y}{9} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.3.1.4.3

Multiply $\frac{2 y}{9}$ by $\frac{2 y}{9}$ .

$\frac{256}{81} - \frac{32 y}{81} - \frac{32 y}{81} + \frac{2 y (2 y)}{9 \cdot 9} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.3.1.4.4

Multiply $2$ by $2$ .

$\frac{256}{81} - \frac{32 y}{81} - \frac{32 y}{81} + \frac{4 y \cdot y}{9 \cdot 9} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.3.1.4.5

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

$\frac{256}{81} - \frac{32 y}{81} - \frac{32 y}{81} + \frac{4 (y \cdot y)}{9 \cdot 9} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.3.1.4.6

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

$\frac{256}{81} - \frac{32 y}{81} - \frac{32 y}{81} + \frac{4 (y \cdot y)}{9 \cdot 9} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.3.1.4.7

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

$\frac{256}{81} - \frac{32 y}{81} - \frac{32 y}{81} + \frac{4 y^{1 + 1}}{9 \cdot 9} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.3.1.4.8

Add $1$ and $1$ .

$\frac{256}{81} - \frac{32 y}{81} - \frac{32 y}{81} + \frac{4 y^{2}}{9 \cdot 9} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.3.1.4.9

Multiply $9$ by $9$ .

$\frac{256}{81} - \frac{32 y}{81} - \frac{32 y}{81} + \frac{4 y^{2}}{81} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

$\frac{256}{81} - \frac{32 y}{81} - \frac{32 y}{81} + \frac{4 y^{2}}{81} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

$\frac{256}{81} - \frac{32 y}{81} - \frac{32 y}{81} + \frac{4 y^{2}}{81} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.3.2

Subtract $\frac{32 y}{81}$ from $- \frac{32 y}{81}$ .

$\frac{256}{81} - 2 \frac{32 y}{81} + \frac{4 y^{2}}{81} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

$\frac{256}{81} - 2 \frac{32 y}{81} + \frac{4 y^{2}}{81} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.4

Simplify each term.

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Step 2.2.1.1.4.1

Multiply $- 2 \frac{32 y}{81}$ .

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Step 2.2.1.1.4.1.1

Combine $- 2$ and $\frac{32 y}{81}$ .

$\frac{256}{81} + \frac{- 2 (32 y)}{81} + \frac{4 y^{2}}{81} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.4.1.2

Multiply $32$ by $- 2$ .

$\frac{256}{81} + \frac{- 64 y}{81} + \frac{4 y^{2}}{81} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

$\frac{256}{81} + \frac{- 64 y}{81} + \frac{4 y^{2}}{81} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.1.4.2

Move the negative in front of the fraction.

$\frac{256}{81} - \frac{64 y}{81} + \frac{4 y^{2}}{81} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

$\frac{256}{81} - \frac{64 y}{81} + \frac{4 y^{2}}{81} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

$\frac{256}{81} - \frac{64 y}{81} + \frac{4 y^{2}}{81} + y^{2} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.2

To write $y^{2}$ as a fraction with a common denominator, multiply by $\frac{81}{81}$ .

$\frac{256}{81} - \frac{64 y}{81} + \frac{4 y^{2}}{81} + y^{2} \cdot \frac{81}{81} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

Combine $y^{2}$ and $\frac{81}{81}$ .

$\frac{256}{81} - \frac{64 y}{81} + \frac{4 y^{2}}{81} + \frac{y^{2} \cdot 81}{81} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$\frac{256}{81} - \frac{64 y}{81} + \frac{4 y^{2} + y^{2} \cdot 81}{81} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.3.3

Combine the numerators over the common denominator.

$\frac{256 - 64 y + 4 y^{2} + y^{2} \cdot 81}{81} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

$\frac{256 - 64 y + 4 y^{2} + y^{2} \cdot 81}{81} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.4

Move $81$ to the left of $y^{2}$ .

$\frac{256 - 64 y + 4 y^{2} + 81 y^{2}}{81} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 2.2.1.5

Add $4 y^{2}$ and $81 y^{2}$ .

$\frac{256 - 64 y + 85 y^{2}}{81} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

$\frac{256 - 64 y + 85 y^{2}}{81} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

$\frac{256 - 64 y + 85 y^{2}}{81} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

$\frac{256 - 64 y + 85 y^{2}}{81} = 9$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3

Solve for $y$ in $\frac{256 - 64 y + 85 y^{2}}{81} = 9$ .

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Step 3.1

Multiply both sides by $81$ .

$\frac{256 - 64 y + 85 y^{2}}{81} \cdot 81 = 9 \cdot 81$

$x = \frac{16}{9} - \frac{2 y}{9}$

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 $\frac{256 - 64 y + 85 y^{2}}{81} \cdot 81$ .

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Step 3.2.1.1.1

Cancel the common factor of $81$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{256 - 64 y + 85 y^{2}}{81} \cdot 81 = 9 \cdot 81$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.2.1.1.1.2

Rewrite the expression.

$256 - 64 y + 85 y^{2} = 9 \cdot 81$

$x = \frac{16}{9} - \frac{2 y}{9}$

$256 - 64 y + 85 y^{2} = 9 \cdot 81$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.2.1.1.2

Simplify the expression.

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Step 3.2.1.1.2.1

Move $256$ .

$- 64 y + 85 y^{2} + 256 = 9 \cdot 81$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.2.1.1.2.2

Reorder $- 64 y$ and $85 y^{2}$ .

$85 y^{2} - 64 y + 256 = 9 \cdot 81$

$x = \frac{16}{9} - \frac{2 y}{9}$

$85 y^{2} - 64 y + 256 = 9 \cdot 81$

$x = \frac{16}{9} - \frac{2 y}{9}$

$85 y^{2} - 64 y + 256 = 9 \cdot 81$

$x = \frac{16}{9} - \frac{2 y}{9}$

$85 y^{2} - 64 y + 256 = 9 \cdot 81$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $9$ by $81$ .

$85 y^{2} - 64 y + 256 = 729$

$x = \frac{16}{9} - \frac{2 y}{9}$

$85 y^{2} - 64 y + 256 = 729$

$x = \frac{16}{9} - \frac{2 y}{9}$

$85 y^{2} - 64 y + 256 = 729$

$x = \frac{16}{9} - \frac{2 y}{9}$

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 $729$ from both sides of the equation.

$85 y^{2} - 64 y + 256 - 729 = 0$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.1.2

Subtract $729$ from $256$ .

$85 y^{2} - 64 y - 473 = 0$

$x = \frac{16}{9} - \frac{2 y}{9}$

$85 y^{2} - 64 y - 473 = 0$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.2

Use the quadratic formula to find the solutions.

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

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.3

Substitute the values $a = 85$ , $b = - 64$ , and $c = - 473$ into the quadratic formula and solve for $y$ .

$\frac{64 \pm \sqrt{{(- 64)}^{2} - 4 \cdot (85 \cdot - 473)}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

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 $- 64$ to the power of $2$ .

$y = \frac{64 \pm \sqrt{4096 - 4 \cdot 85 \cdot - 473}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.4.1.2

Multiply $- 4 \cdot 85 \cdot - 473$ .

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Step 3.3.4.1.2.1

Multiply $- 4$ by $85$ .

$y = \frac{64 \pm \sqrt{4096 - 340 \cdot - 473}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.4.1.2.2

Multiply $- 340$ by $- 473$ .

$y = \frac{64 \pm \sqrt{4096 + 160820}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

$y = \frac{64 \pm \sqrt{4096 + 160820}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.4.1.3

Add $4096$ and $160820$ .

$y = \frac{64 \pm \sqrt{164916}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.4.1.4

Rewrite $164916$ as $18^{2} \cdot 509$ .

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Step 3.3.4.1.4.1

Factor $324$ out of $164916$ .

$y = \frac{64 \pm \sqrt{324 (509)}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.4.1.4.2

Rewrite $324$ as $18^{2}$ .

$y = \frac{64 \pm \sqrt{18^{2} \cdot 509}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

$y = \frac{64 \pm \sqrt{18^{2} \cdot 509}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.4.1.5

Pull terms out from under the radical.

$y = \frac{64 \pm 18 \sqrt{509}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

$y = \frac{64 \pm 18 \sqrt{509}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.4.2

Multiply $2$ by $85$ .

$y = \frac{64 \pm 18 \sqrt{509}}{170}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.4.3

Simplify $\frac{64 \pm 18 \sqrt{509}}{170}$ .

$y = \frac{32 \pm 9 \sqrt{509}}{85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

$y = \frac{32 \pm 9 \sqrt{509}}{85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

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 $- 64$ to the power of $2$ .

$y = \frac{64 \pm \sqrt{4096 - 4 \cdot 85 \cdot - 473}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.5.1.2

Multiply $- 4 \cdot 85 \cdot - 473$ .

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Step 3.3.5.1.2.1

Multiply $- 4$ by $85$ .

$y = \frac{64 \pm \sqrt{4096 - 340 \cdot - 473}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.5.1.2.2

Multiply $- 340$ by $- 473$ .

$y = \frac{64 \pm \sqrt{4096 + 160820}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

$y = \frac{64 \pm \sqrt{4096 + 160820}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.5.1.3

Add $4096$ and $160820$ .

$y = \frac{64 \pm \sqrt{164916}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.5.1.4

Rewrite $164916$ as $18^{2} \cdot 509$ .

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Step 3.3.5.1.4.1

Factor $324$ out of $164916$ .

$y = \frac{64 \pm \sqrt{324 (509)}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.5.1.4.2

Rewrite $324$ as $18^{2}$ .

$y = \frac{64 \pm \sqrt{18^{2} \cdot 509}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

$y = \frac{64 \pm \sqrt{18^{2} \cdot 509}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.5.1.5

Pull terms out from under the radical.

$y = \frac{64 \pm 18 \sqrt{509}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

$y = \frac{64 \pm 18 \sqrt{509}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.5.2

Multiply $2$ by $85$ .

$y = \frac{64 \pm 18 \sqrt{509}}{170}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.5.3

Simplify $\frac{64 \pm 18 \sqrt{509}}{170}$ .

$y = \frac{32 \pm 9 \sqrt{509}}{85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.5.4

Change the $\pm$ to $+$ .

$y = \frac{32 + 9 \sqrt{509}}{85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

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 $- 64$ to the power of $2$ .

$y = \frac{64 \pm \sqrt{4096 - 4 \cdot 85 \cdot - 473}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.6.1.2

Multiply $- 4 \cdot 85 \cdot - 473$ .

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Step 3.3.6.1.2.1

Multiply $- 4$ by $85$ .

$y = \frac{64 \pm \sqrt{4096 - 340 \cdot - 473}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.6.1.2.2

Multiply $- 340$ by $- 473$ .

$y = \frac{64 \pm \sqrt{4096 + 160820}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

$y = \frac{64 \pm \sqrt{4096 + 160820}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.6.1.3

Add $4096$ and $160820$ .

$y = \frac{64 \pm \sqrt{164916}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.6.1.4

Rewrite $164916$ as $18^{2} \cdot 509$ .

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Step 3.3.6.1.4.1

Factor $324$ out of $164916$ .

$y = \frac{64 \pm \sqrt{324 (509)}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.6.1.4.2

Rewrite $324$ as $18^{2}$ .

$y = \frac{64 \pm \sqrt{18^{2} \cdot 509}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

$y = \frac{64 \pm \sqrt{18^{2} \cdot 509}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.6.1.5

Pull terms out from under the radical.

$y = \frac{64 \pm 18 \sqrt{509}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

$y = \frac{64 \pm 18 \sqrt{509}}{2 \cdot 85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.6.2

Multiply $2$ by $85$ .

$y = \frac{64 \pm 18 \sqrt{509}}{170}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.6.3

Simplify $\frac{64 \pm 18 \sqrt{509}}{170}$ .

$y = \frac{32 \pm 9 \sqrt{509}}{85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.6.4

Change the $\pm$ to $-$ .

$y = \frac{32 - 9 \sqrt{509}}{85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 3.3.7

The final answer is the combination of both solutions.

$y = \frac{32 + 9 \sqrt{509}}{85}, \frac{32 - 9 \sqrt{509}}{85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

$y = \frac{32 + 9 \sqrt{509}}{85}, \frac{32 - 9 \sqrt{509}}{85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

$y = \frac{32 + 9 \sqrt{509}}{85}, \frac{32 - 9 \sqrt{509}}{85}$

$x = \frac{16}{9} - \frac{2 y}{9}$

Step 4

Replace all occurrences of $y$ with $\frac{32 + 9 \sqrt{509}}{85}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{16}{9} - \frac{2 y}{9}$ with $\frac{32 + 9 \sqrt{509}}{85}$ .

$x = \frac{16}{9} - \frac{2 (\frac{32 + 9 \sqrt{509}}{85})}{9}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{16}{9} - \frac{2 (\frac{32 + 9 \sqrt{509}}{85})}{9}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{16 - 2 \frac{32 + 9 \sqrt{509}}{85}}{9}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

Step 4.2.1.2

Simplify each term.

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Step 4.2.1.2.1

Combine $- 2$ and $\frac{32 + 9 \sqrt{509}}{85}$ .

$x = \frac{16 + \frac{- 2 (32 + 9 \sqrt{509})}{85}}{9}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

Step 4.2.1.2.2

Move the negative in front of the fraction.

$x = \frac{16 - \frac{2 (32 + 9 \sqrt{509})}{85}}{9}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

$x = \frac{16 - \frac{2 (32 + 9 \sqrt{509})}{85}}{9}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

Step 4.2.1.3

To write $16$ as a fraction with a common denominator, multiply by $\frac{85}{85}$ .

$x = \frac{16 \cdot \frac{85}{85} - \frac{2 (32 + 9 \sqrt{509})}{85}}{9}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

Step 4.2.1.4

Combine fractions.

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Step 4.2.1.4.1

Combine $16$ and $\frac{85}{85}$ .

$x = \frac{\frac{16 \cdot 85}{85} - \frac{2 (32 + 9 \sqrt{509})}{85}}{9}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

Step 4.2.1.4.2

Combine the numerators over the common denominator.

$x = \frac{\frac{16 \cdot 85 - 2 (32 + 9 \sqrt{509})}{85}}{9}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

$x = \frac{\frac{16 \cdot 85 - 2 (32 + 9 \sqrt{509})}{85}}{9}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $16$ by $85$ .

$x = \frac{\frac{1360 - 2 (32 + 9 \sqrt{509})}{85}}{9}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

Step 4.2.1.5.2

Apply the distributive property.

$x = \frac{\frac{1360 - 2 \cdot 32 - 2 (9 \sqrt{509})}{85}}{9}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

Step 4.2.1.5.3

Multiply $- 2$ by $32$ .

$x = \frac{\frac{1360 - 64 - 2 (9 \sqrt{509})}{85}}{9}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

Step 4.2.1.5.4

Multiply $9$ by $- 2$ .

$x = \frac{\frac{1360 - 64 - 18 \sqrt{509}}{85}}{9}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

Step 4.2.1.5.5

Subtract $64$ from $1360$ .

$x = \frac{\frac{1296 - 18 \sqrt{509}}{85}}{9}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

$x = \frac{\frac{1296 - 18 \sqrt{509}}{85}}{9}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

Step 4.2.1.6

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{1296 - 18 \sqrt{509}}{85} \cdot \frac{1}{9}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

Step 4.2.1.7

Multiply $\frac{1296 - 18 \sqrt{509}}{85} \cdot \frac{1}{9}$ .

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Step 4.2.1.7.1

Multiply $\frac{1296 - 18 \sqrt{509}}{85}$ by $\frac{1}{9}$ .

$x = \frac{1296 - 18 \sqrt{509}}{85 \cdot 9}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

Step 4.2.1.7.2

Multiply $85$ by $9$ .

$x = \frac{1296 - 18 \sqrt{509}}{765}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

$x = \frac{1296 - 18 \sqrt{509}}{765}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

Step 4.2.1.8

Cancel the common factor of $1296 - 18 \sqrt{509}$ and $765$ .

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Step 4.2.1.8.1

Factor $9$ out of $1296$ .

$x = \frac{9 (144) - 18 \sqrt{509}}{765}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

Step 4.2.1.8.2

Factor $9$ out of $- 18 \sqrt{509}$ .

$x = \frac{9 (144) + 9 (- 2 \sqrt{509})}{765}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

Step 4.2.1.8.3

Factor $9$ out of $9 (144) + 9 (- 2 \sqrt{509})$ .

$x = \frac{9 (144 - 2 \sqrt{509})}{765}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

Step 4.2.1.8.4

Cancel the common factors.

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Step 4.2.1.8.4.1

Factor $9$ out of $765$ .

$x = \frac{9 (144 - 2 \sqrt{509})}{9 \cdot 85}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

Step 4.2.1.8.4.2

Cancel the common factor.

$x = \frac{9 (144 - 2 \sqrt{509})}{9 \cdot 85}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

Step 4.2.1.8.4.3

Rewrite the expression.

$x = \frac{144 - 2 \sqrt{509}}{85}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

$x = \frac{144 - 2 \sqrt{509}}{85}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

$x = \frac{144 - 2 \sqrt{509}}{85}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

$x = \frac{144 - 2 \sqrt{509}}{85}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

$x = \frac{144 - 2 \sqrt{509}}{85}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

$x = \frac{144 - 2 \sqrt{509}}{85}$

$y = \frac{32 + 9 \sqrt{509}}{85}$

Step 5

Replace all occurrences of $y$ with $\frac{32 - 9 \sqrt{509}}{85}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = \frac{16}{9} - \frac{2 y}{9}$ with $\frac{32 - 9 \sqrt{509}}{85}$ .

$x = \frac{16}{9} - \frac{2 (\frac{32 - 9 \sqrt{509}}{85})}{9}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{16}{9} - \frac{2 (\frac{32 - 9 \sqrt{509}}{85})}{9}$ .

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Step 5.2.1.1

Combine the numerators over the common denominator.

$x = \frac{16 - 2 \frac{32 - 9 \sqrt{509}}{85}}{9}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

Step 5.2.1.2

Simplify each term.

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Step 5.2.1.2.1

Combine $- 2$ and $\frac{32 - 9 \sqrt{509}}{85}$ .

$x = \frac{16 + \frac{- 2 (32 - 9 \sqrt{509})}{85}}{9}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

Step 5.2.1.2.2

Move the negative in front of the fraction.

$x = \frac{16 - \frac{2 (32 - 9 \sqrt{509})}{85}}{9}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

$x = \frac{16 - \frac{2 (32 - 9 \sqrt{509})}{85}}{9}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

Step 5.2.1.3

To write $16$ as a fraction with a common denominator, multiply by $\frac{85}{85}$ .

$x = \frac{16 \cdot \frac{85}{85} - \frac{2 (32 - 9 \sqrt{509})}{85}}{9}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

Step 5.2.1.4

Combine fractions.

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Step 5.2.1.4.1

Combine $16$ and $\frac{85}{85}$ .

$x = \frac{\frac{16 \cdot 85}{85} - \frac{2 (32 - 9 \sqrt{509})}{85}}{9}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

Step 5.2.1.4.2

Combine the numerators over the common denominator.

$x = \frac{\frac{16 \cdot 85 - 2 (32 - 9 \sqrt{509})}{85}}{9}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

$x = \frac{\frac{16 \cdot 85 - 2 (32 - 9 \sqrt{509})}{85}}{9}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

Step 5.2.1.5

Simplify the numerator.

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Step 5.2.1.5.1

Multiply $16$ by $85$ .

$x = \frac{\frac{1360 - 2 (32 - 9 \sqrt{509})}{85}}{9}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

Step 5.2.1.5.2

Apply the distributive property.

$x = \frac{\frac{1360 - 2 \cdot 32 - 2 (- 9 \sqrt{509})}{85}}{9}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

Step 5.2.1.5.3

Multiply $- 2$ by $32$ .

$x = \frac{\frac{1360 - 64 - 2 (- 9 \sqrt{509})}{85}}{9}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

Step 5.2.1.5.4

Multiply $- 9$ by $- 2$ .

$x = \frac{\frac{1360 - 64 + 18 \sqrt{509}}{85}}{9}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

Step 5.2.1.5.5

Subtract $64$ from $1360$ .

$x = \frac{\frac{1296 + 18 \sqrt{509}}{85}}{9}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

$x = \frac{\frac{1296 + 18 \sqrt{509}}{85}}{9}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

Step 5.2.1.6

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{1296 + 18 \sqrt{509}}{85} \cdot \frac{1}{9}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

Step 5.2.1.7

Multiply $\frac{1296 + 18 \sqrt{509}}{85} \cdot \frac{1}{9}$ .

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Step 5.2.1.7.1

Multiply $\frac{1296 + 18 \sqrt{509}}{85}$ by $\frac{1}{9}$ .

$x = \frac{1296 + 18 \sqrt{509}}{85 \cdot 9}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

Step 5.2.1.7.2

Multiply $85$ by $9$ .

$x = \frac{1296 + 18 \sqrt{509}}{765}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

$x = \frac{1296 + 18 \sqrt{509}}{765}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

Step 5.2.1.8

Cancel the common factor of $1296 + 18 \sqrt{509}$ and $765$ .

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Step 5.2.1.8.1

Factor $9$ out of $1296$ .

$x = \frac{9 (144) + 18 \sqrt{509}}{765}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

Step 5.2.1.8.2

Factor $9$ out of $18 \sqrt{509}$ .

$x = \frac{9 (144) + 9 (2 \sqrt{509})}{765}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

Step 5.2.1.8.3

Factor $9$ out of $9 (144) + 9 (2 \sqrt{509})$ .

$x = \frac{9 (144 + 2 \sqrt{509})}{765}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

Step 5.2.1.8.4

Cancel the common factors.

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Step 5.2.1.8.4.1

Factor $9$ out of $765$ .

$x = \frac{9 (144 + 2 \sqrt{509})}{9 \cdot 85}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

Step 5.2.1.8.4.2

Cancel the common factor.

$x = \frac{9 (144 + 2 \sqrt{509})}{9 \cdot 85}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

Step 5.2.1.8.4.3

Rewrite the expression.

$x = \frac{144 + 2 \sqrt{509}}{85}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

$x = \frac{144 + 2 \sqrt{509}}{85}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

$x = \frac{144 + 2 \sqrt{509}}{85}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

$x = \frac{144 + 2 \sqrt{509}}{85}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

$x = \frac{144 + 2 \sqrt{509}}{85}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

$x = \frac{144 + 2 \sqrt{509}}{85}$

$y = \frac{32 - 9 \sqrt{509}}{85}$

Step 6

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

$(\frac{144 - 2 \sqrt{509}}{85}, \frac{32 + 9 \sqrt{509}}{85})$

$(\frac{144 + 2 \sqrt{509}}{85}, \frac{32 - 9 \sqrt{509}}{85})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(\frac{144 - 2 \sqrt{509}}{85}, \frac{32 + 9 \sqrt{509}}{85}), (\frac{144 + 2 \sqrt{509}}{85}, \frac{32 - 9 \sqrt{509}}{85})$

Equation Form:

$x = \frac{144 - 2 \sqrt{509}}{85}, y = \frac{32 + 9 \sqrt{509}}{85}$

$x = \frac{144 + 2 \sqrt{509}}{85}, y = \frac{32 - 9 \sqrt{509}}{85}$

Step 8