Solve the System of Equations x^2+y^2=9 9x+2y=16

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

Step 1

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

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

$9 x = 16 - 2 y$

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

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

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

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

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

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

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

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

Divide $x$ by $1$ .

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

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

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

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

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

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

Simplify the right side.

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.

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

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

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

Simplify the left side.

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

Simplify each term.

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

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

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

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

Apply the distributive property.

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

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

Apply the distributive property.

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

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

Apply the distributive property.

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

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

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

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

Simplify and combine like terms.

Simplify each term.

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

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

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

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

Multiply $16$ by $16$ .

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

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

Multiply $9$ by $9$ .

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

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

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

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

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

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

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

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

Multiply $2$ by $16$ .

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

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

Multiply $9$ by $9$ .

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

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

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

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

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

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

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

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

Multiply $2$ by $16$ .

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

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

Multiply $9$ by $9$ .

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

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

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

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

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

Multiply $- 1$ by $- 1$ .

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

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

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

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

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

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

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

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

Multiply $2$ by $2$ .

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

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

Multiply $9$ by $9$ .

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

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

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

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

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

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

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

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

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

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

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

Simplify each term.

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

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

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

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

Multiply $32$ by $- 2$ .

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

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

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

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

Move the negative in front of the fraction.

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

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

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

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

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

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

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

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

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

Simplify terms.

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

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

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

Combine the numerators over the common denominator.

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

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

Combine the numerators over the common denominator.

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

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

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

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

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

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

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

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

Multiply both sides by $81$ .

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

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

Simplify.

Simplify the left side.

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

Cancel the common factor of $81$ .

Cancel the common factor.

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

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

Rewrite the expression.

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

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

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

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

Simplify the expression.

Move $256$ .

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

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

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

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

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

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

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

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

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

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

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

Simplify the right side.

Multiply $9$ by $81$ .

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

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

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

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

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

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

Solve for $y$ .

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

Subtract $729$ from both sides of the equation.

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

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

Subtract $729$ from $256$ .

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

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

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

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

Use the quadratic formula to find the solutions.

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

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

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

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

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

Simplify.

Simplify the numerator.

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

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

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

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

Multiply $- 4$ by $85$ .

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

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

Multiply $- 340$ by $- 473$ .

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

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

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

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

Add $4096$ and $160820$ .

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

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

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

Factor $324$ out of $164916$ .

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

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

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

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

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

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

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

Pull terms out from under the radical.

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

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

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

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

Multiply $2$ by $85$ .

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

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

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

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

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

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

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

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

Simplify the numerator.

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

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

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

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

Multiply $- 4$ by $85$ .

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

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

Multiply $- 340$ by $- 473$ .

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

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

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

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

Add $4096$ and $160820$ .

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

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

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

Factor $324$ out of $164916$ .

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

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

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

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

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

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

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

Pull terms out from under the radical.

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

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

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

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

Multiply $2$ by $85$ .

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

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

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

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

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

Change the $\pm$ to $+$ .

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

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

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

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

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

Simplify the numerator.

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

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

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

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

Multiply $- 4$ by $85$ .

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

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

Multiply $- 340$ by $- 473$ .

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

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

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

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

Add $4096$ and $160820$ .

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

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

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

Factor $324$ out of $164916$ .

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

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

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

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

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

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

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

Pull terms out from under the radical.

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

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

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

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

Multiply $2$ by $85$ .

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

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

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

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

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

Change the $\pm$ to $-$ .

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

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

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

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

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.

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

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

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

Simplify the right side.

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

Combine the numerators over the common denominator.

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

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

Simplify each term.

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

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

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

Move the negative in front of the fraction.

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

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

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

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

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

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

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

Combine fractions.

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

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

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

Combine the numerators over the common denominator.

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

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

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

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

Simplify the numerator.

Multiply $16$ by $85$ .

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

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

Apply the distributive property.

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

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

Multiply $- 2$ by $32$ .

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

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

Multiply $9$ by $- 2$ .

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

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

Subtract $64$ from $1360$ .

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

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

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

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

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

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

Multiply $85$ by $9$ .

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

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

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

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

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

Factor $9$ out of $1296$ .

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

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

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

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

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

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

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

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

Cancel the common factors.

Factor $9$ out of $765$ .

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

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

Cancel the common factor.

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

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

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.

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

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

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

Simplify the right side.

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

Combine the numerators over the common denominator.

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

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

Simplify each term.

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

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

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

Move the negative in front of the fraction.

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

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

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

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

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

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

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

Combine fractions.

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

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

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

Combine the numerators over the common denominator.

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

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

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

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

Simplify the numerator.

Multiply $16$ by $85$ .

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

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

Apply the distributive property.

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

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

Multiply $- 2$ by $32$ .

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

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

Multiply $- 9$ by $- 2$ .

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

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

Subtract $64$ from $1360$ .

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

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

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

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

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

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

Multiply $85$ by $9$ .

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

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

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

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

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

Factor $9$ out of $1296$ .

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

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

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

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

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

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

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

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

Cancel the common factors.

Factor $9$ out of $765$ .

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

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

Cancel the common factor.

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

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

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

Solve the System of Equations x^2+y^2=9 9x+2y=16

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