Solve by Substitution x^2+y^2=16 , y^2-2x^2=10

$x^{2} + y^{2} = 16$ , $y^{2} - 2 x^{2} = 10$

Step 1

Solve for $x$ in $x^{2} + y^{2} = 16$ .

Subtract $y^{2}$ from both sides of the equation.

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

$y^{2} - 2 x^{2} = 10$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{16 - y^{2}}$

$y^{2} - 2 x^{2} = 10$

Simplify $\pm \sqrt{16 - y^{2}}$ .

Rewrite $16$ as $4^{2}$ .

$x = \pm \sqrt{4^{2} - y^{2}}$

$y^{2} - 2 x^{2} = 10$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = y$ .

$x = \pm \sqrt{(4 + y) (4 - y)}$

$y^{2} - 2 x^{2} = 10$

$x = \pm \sqrt{(4 + y) (4 - y)}$

$y^{2} - 2 x^{2} = 10$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{(4 + y) (4 - y)}$

$y^{2} - 2 x^{2} = 10$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{(4 + y) (4 - y)}$

$y^{2} - 2 x^{2} = 10$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{(4 + y) (4 - y)}$

$x = - \sqrt{(4 + y) (4 - y)}$

$y^{2} - 2 x^{2} = 10$

$x = \sqrt{(4 + y) (4 - y)}$

$x = - \sqrt{(4 + y) (4 - y)}$

$y^{2} - 2 x^{2} = 10$

$x = \sqrt{(4 + y) (4 - y)}$

$x = - \sqrt{(4 + y) (4 - y)}$

$y^{2} - 2 x^{2} = 10$

Step 2

Solve the system $x = \sqrt{(4 + y) (4 - y)}, y^{2} - 2 x^{2} = 10$ .

Replace all occurrences of $x$ with $\sqrt{(4 + y) (4 - y)}$ in each equation.

Replace all occurrences of $x$ in $y^{2} - 2 x^{2} = 10$ with $\sqrt{(4 + y) (4 - y)}$ .

$y^{2} - 2 {(\sqrt{(4 + y) (4 - y)})}^{2} = 10$

$x = \sqrt{(4 + y) (4 - y)}$

Simplify the left side.

Simplify $y^{2} - 2 {(\sqrt{(4 + y) (4 - y)})}^{2}$ .

Simplify each term.

Rewrite ${\sqrt{(4 + y) (4 - y)}}^{2}$ as $(4 + y) (4 - y)$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(4 + y) (4 - y)}$ as ${((4 + y) (4 - y))}^{\frac{1}{2}}$ .

$y^{2} - 2 {({((4 + y) (4 - y))}^{\frac{1}{2}})}^{2} = 10$

$x = \sqrt{(4 + y) (4 - y)}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y^{2} - 2 {((4 + y) (4 - y))}^{\frac{1}{2} \cdot 2} = 10$

$x = \sqrt{(4 + y) (4 - y)}$

Combine $\frac{1}{2}$ and $2$ .

$y^{2} - 2 {((4 + y) (4 - y))}^{\frac{2}{2}} = 10$

$x = \sqrt{(4 + y) (4 - y)}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y^{2} - 2 {((4 + y) (4 - y))}^{\frac{2}{2}} = 10$

$x = \sqrt{(4 + y) (4 - y)}$

Rewrite the expression.

$y^{2} - 2 ((4 + y) (4 - y)) = 10$

$x = \sqrt{(4 + y) (4 - y)}$

$y^{2} - 2 ((4 + y) (4 - y)) = 10$

$x = \sqrt{(4 + y) (4 - y)}$

Simplify.

$y^{2} - 2 ((4 + y) (4 - y)) = 10$

$x = \sqrt{(4 + y) (4 - y)}$

$y^{2} - 2 ((4 + y) (4 - y)) = 10$

$x = \sqrt{(4 + y) (4 - y)}$

Expand $(4 + y) (4 - y)$ using the FOIL Method.

Apply the distributive property.

$y^{2} - 2 (4 (4 - y) + y (4 - y)) = 10$

$x = \sqrt{(4 + y) (4 - y)}$

Apply the distributive property.

$y^{2} - 2 (4 \cdot 4 + 4 (- y) + y (4 - y)) = 10$

$x = \sqrt{(4 + y) (4 - y)}$

Apply the distributive property.

$y^{2} - 2 (4 \cdot 4 + 4 (- y) + y \cdot 4 + y (- y)) = 10$

$x = \sqrt{(4 + y) (4 - y)}$

$y^{2} - 2 (4 \cdot 4 + 4 (- y) + y \cdot 4 + y (- y)) = 10$

$x = \sqrt{(4 + y) (4 - y)}$

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

$y^{2} - 2 (16 + 4 (- y) + y \cdot 4 + y (- y)) = 10$

$x = \sqrt{(4 + y) (4 - y)}$

Multiply $- 1$ by $4$ .

$y^{2} - 2 (16 - 4 y + y \cdot 4 + y (- y)) = 10$

$x = \sqrt{(4 + y) (4 - y)}$

Move $4$ to the left of $y$ .

$y^{2} - 2 (16 - 4 y + 4 \cdot y + y (- y)) = 10$

$x = \sqrt{(4 + y) (4 - y)}$

Rewrite using the commutative property of multiplication.

$y^{2} - 2 (16 - 4 y + 4 y - y \cdot y) = 10$

$x = \sqrt{(4 + y) (4 - y)}$

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

Move $y$ .

$y^{2} - 2 (16 - 4 y + 4 y - (y \cdot y)) = 10$

$x = \sqrt{(4 + y) (4 - y)}$

Multiply $y$ by $y$ .

$y^{2} - 2 (16 - 4 y + 4 y - y^{2}) = 10$

$x = \sqrt{(4 + y) (4 - y)}$

$y^{2} - 2 (16 - 4 y + 4 y - y^{2}) = 10$

$x = \sqrt{(4 + y) (4 - y)}$

$y^{2} - 2 (16 - 4 y + 4 y - y^{2}) = 10$

$x = \sqrt{(4 + y) (4 - y)}$

Add $- 4 y$ and $4 y$ .

$y^{2} - 2 (16 + 0 - y^{2}) = 10$

$x = \sqrt{(4 + y) (4 - y)}$

Add $16$ and $0$ .

$y^{2} - 2 (16 - y^{2}) = 10$

$x = \sqrt{(4 + y) (4 - y)}$

$y^{2} - 2 (16 - y^{2}) = 10$

$x = \sqrt{(4 + y) (4 - y)}$

Apply the distributive property.

$y^{2} - 2 \cdot 16 - 2 (- y^{2}) = 10$

$x = \sqrt{(4 + y) (4 - y)}$

Multiply $- 2$ by $16$ .

$y^{2} - 32 - 2 (- y^{2}) = 10$

$x = \sqrt{(4 + y) (4 - y)}$

Multiply $- 1$ by $- 2$ .

$y^{2} - 32 + 2 y^{2} = 10$

$x = \sqrt{(4 + y) (4 - y)}$

$y^{2} - 32 + 2 y^{2} = 10$

$x = \sqrt{(4 + y) (4 - y)}$

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

$3 y^{2} - 32 = 10$

$x = \sqrt{(4 + y) (4 - y)}$

$3 y^{2} - 32 = 10$

$x = \sqrt{(4 + y) (4 - y)}$

$3 y^{2} - 32 = 10$

$x = \sqrt{(4 + y) (4 - y)}$

$3 y^{2} - 32 = 10$

$x = \sqrt{(4 + y) (4 - y)}$

Solve for $y$ in $3 y^{2} - 32 = 10$ .

Move all terms not containing $y$ to the right side of the equation.

Add $32$ to both sides of the equation.

$3 y^{2} = 10 + 32$

$x = \sqrt{(4 + y) (4 - y)}$

Add $10$ and $32$ .

$3 y^{2} = 42$

$x = \sqrt{(4 + y) (4 - y)}$

$3 y^{2} = 42$

$x = \sqrt{(4 + y) (4 - y)}$

Divide each term in $3 y^{2} = 42$ by $3$ and simplify.

Divide each term in $3 y^{2} = 42$ by $3$ .

$\frac{3 y^{2}}{3} = \frac{42}{3}$

$x = \sqrt{(4 + y) (4 - y)}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y^{2}}{3} = \frac{42}{3}$

$x = \sqrt{(4 + y) (4 - y)}$

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{42}{3}$

$x = \sqrt{(4 + y) (4 - y)}$

$y^{2} = \frac{42}{3}$

$x = \sqrt{(4 + y) (4 - y)}$

$y^{2} = \frac{42}{3}$

$x = \sqrt{(4 + y) (4 - y)}$

Simplify the right side.

Divide $42$ by $3$ .

$y^{2} = 14$

$x = \sqrt{(4 + y) (4 - y)}$

$y^{2} = 14$

$x = \sqrt{(4 + y) (4 - y)}$

$y^{2} = 14$

$x = \sqrt{(4 + y) (4 - y)}$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{14}$

$x = \sqrt{(4 + y) (4 - y)}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{14}$

$x = \sqrt{(4 + y) (4 - y)}$

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{14}$

$x = \sqrt{(4 + y) (4 - y)}$

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{14}, - \sqrt{14}$

$x = \sqrt{(4 + y) (4 - y)}$

$y = \sqrt{14}, - \sqrt{14}$

$x = \sqrt{(4 + y) (4 - y)}$

$y = \sqrt{14}, - \sqrt{14}$

$x = \sqrt{(4 + y) (4 - y)}$

Replace all occurrences of $y$ with $\sqrt{14}$ in each equation.

Replace all occurrences of $y$ in $x = \sqrt{(4 + y) (4 - y)}$ with $\sqrt{14}$ .

$x = \sqrt{(4 + \sqrt{14}) (4 - (\sqrt{14}))}$

$y = \sqrt{14}$

Simplify $x = \sqrt{(4 + \sqrt{14}) (4 - (\sqrt{14}))}$ .

Simplify the left side.

Remove parentheses.

$x = \sqrt{(4 + \sqrt{14}) (4 - (\sqrt{14}))}$

$y = \sqrt{14}$

$x = \sqrt{(4 + \sqrt{14}) (4 - (\sqrt{14}))}$

$y = \sqrt{14}$

Simplify the right side.

Simplify $\sqrt{(4 + \sqrt{14}) (4 - (\sqrt{14}))}$ .

Expand $(4 + \sqrt{14}) (4 - \sqrt{14})$ using the FOIL Method.

Apply the distributive property.

$x = \sqrt{4 (4 - \sqrt{14}) + \sqrt{14} (4 - \sqrt{14})}$

$y = \sqrt{14}$

Apply the distributive property.

$x = \sqrt{4 \cdot 4 + 4 (- \sqrt{14}) + \sqrt{14} (4 - \sqrt{14})}$

$y = \sqrt{14}$

Apply the distributive property.

$x = \sqrt{4 \cdot 4 + 4 (- \sqrt{14}) + \sqrt{14} \cdot 4 + \sqrt{14} (- \sqrt{14})}$

$y = \sqrt{14}$

$x = \sqrt{4 \cdot 4 + 4 (- \sqrt{14}) + \sqrt{14} \cdot 4 + \sqrt{14} (- \sqrt{14})}$

$y = \sqrt{14}$

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

$x = \sqrt{16 + 4 (- \sqrt{14}) + \sqrt{14} \cdot 4 + \sqrt{14} (- \sqrt{14})}$

$y = \sqrt{14}$

Multiply $- 1$ by $4$ .

$x = \sqrt{16 - 4 \sqrt{14} + \sqrt{14} \cdot 4 + \sqrt{14} (- \sqrt{14})}$

$y = \sqrt{14}$

Move $4$ to the left of $\sqrt{14}$ .

$x = \sqrt{16 - 4 \sqrt{14} + 4 \cdot \sqrt{14} + \sqrt{14} (- \sqrt{14})}$

$y = \sqrt{14}$

Multiply $\sqrt{14} (- \sqrt{14})$ .

Raise $\sqrt{14}$ to the power of $1$ .

$x = \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - (\sqrt{14} \sqrt{14})}$

$y = \sqrt{14}$

Raise $\sqrt{14}$ to the power of $1$ .

$x = \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - (\sqrt{14} \sqrt{14})}$

$y = \sqrt{14}$

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

$x = \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - {\sqrt{14}}^{1 + 1}}$

$y = \sqrt{14}$

Add $1$ and $1$ .

$x = \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - {\sqrt{14}}^{2}}$

$y = \sqrt{14}$

$x = \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - {\sqrt{14}}^{2}}$

$y = \sqrt{14}$

Rewrite ${\sqrt{14}}^{2}$ as $14$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{14}$ as $14^{\frac{1}{2}}$ .

$x = \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - {(14^{\frac{1}{2}})}^{2}}$

$y = \sqrt{14}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - 14^{\frac{1}{2} \cdot 2}}$

$y = \sqrt{14}$

Combine $\frac{1}{2}$ and $2$ .

$x = \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - 14^{\frac{2}{2}}}$

$y = \sqrt{14}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - 14^{\frac{2}{2}}}$

$y = \sqrt{14}$

Rewrite the expression.

$x = \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - 14}$

$y = \sqrt{14}$

$x = \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - 14}$

$y = \sqrt{14}$

Evaluate the exponent.

$x = \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - 1 \cdot 14}$

$y = \sqrt{14}$

$x = \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - 1 \cdot 14}$

$y = \sqrt{14}$

Multiply $- 1$ by $14$ .

$x = \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - 14}$

$y = \sqrt{14}$

$x = \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - 14}$

$y = \sqrt{14}$

Subtract $14$ from $16$ .

$x = \sqrt{2 - 4 \sqrt{14} + 4 \sqrt{14}}$

$y = \sqrt{14}$

Add $- 4 \sqrt{14}$ and $4 \sqrt{14}$ .

$x = \sqrt{2 + 0}$

$y = \sqrt{14}$

Add $2$ and $0$ .

$x = \sqrt{2}$

$y = \sqrt{14}$

$x = \sqrt{2}$

$y = \sqrt{14}$

$x = \sqrt{2}$

$y = \sqrt{14}$

$x = \sqrt{2}$

$y = \sqrt{14}$

$x = \sqrt{2}$

$y = \sqrt{14}$

$x = \sqrt{2}$

$y = \sqrt{14}$

Replace all occurrences of $y$ with $- \sqrt{14}$ in each equation.

Replace all occurrences of $y$ in $x = \sqrt{(4 + y) (4 - y)}$ with $- \sqrt{14}$ .

$x = \sqrt{(4 - \sqrt{14}) (4 - (- \sqrt{14}))}$

$y = - \sqrt{14}$

Simplify $x = \sqrt{(4 - \sqrt{14}) (4 - (- \sqrt{14}))}$ .

Simplify the left side.

Remove parentheses.

$x = \sqrt{(4 - \sqrt{14}) (4 - (- \sqrt{14}))}$

$y = - \sqrt{14}$

$x = \sqrt{(4 - \sqrt{14}) (4 - (- \sqrt{14}))}$

$y = - \sqrt{14}$

Simplify the right side.

Simplify $\sqrt{(4 - \sqrt{14}) (4 - (- \sqrt{14}))}$ .

Multiply $- - \sqrt{14}$ .

Multiply $- 1$ by $- 1$ .

$x = \sqrt{(4 - \sqrt{14}) (4 + 1 \sqrt{14})}$

$y = - \sqrt{14}$

Multiply $\sqrt{14}$ by $1$ .

$x = \sqrt{(4 - \sqrt{14}) (4 + \sqrt{14})}$

$y = - \sqrt{14}$

$x = \sqrt{(4 - \sqrt{14}) (4 + \sqrt{14})}$

$y = - \sqrt{14}$

Expand $(4 - \sqrt{14}) (4 + \sqrt{14})$ using the FOIL Method.

Apply the distributive property.

$x = \sqrt{4 (4 + \sqrt{14}) - \sqrt{14} (4 + \sqrt{14})}$

$y = - \sqrt{14}$

Apply the distributive property.

$x = \sqrt{4 \cdot 4 + 4 \sqrt{14} - \sqrt{14} (4 + \sqrt{14})}$

$y = - \sqrt{14}$

Apply the distributive property.

$x = \sqrt{4 \cdot 4 + 4 \sqrt{14} - \sqrt{14} \cdot 4 - \sqrt{14} \sqrt{14}}$

$y = - \sqrt{14}$

$x = \sqrt{4 \cdot 4 + 4 \sqrt{14} - \sqrt{14} \cdot 4 - \sqrt{14} \sqrt{14}}$

$y = - \sqrt{14}$

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

$x = \sqrt{16 + 4 \sqrt{14} - \sqrt{14} \cdot 4 - \sqrt{14} \sqrt{14}}$

$y = - \sqrt{14}$

Multiply $4$ by $- 1$ .

$x = \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - \sqrt{14} \sqrt{14}}$

$y = - \sqrt{14}$

Multiply $- \sqrt{14} \sqrt{14}$ .

Raise $\sqrt{14}$ to the power of $1$ .

$x = \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - (\sqrt{14} \sqrt{14})}$

$y = - \sqrt{14}$

Raise $\sqrt{14}$ to the power of $1$ .

$x = \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - (\sqrt{14} \sqrt{14})}$

$y = - \sqrt{14}$

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

$x = \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - {\sqrt{14}}^{1 + 1}}$

$y = - \sqrt{14}$

Add $1$ and $1$ .

$x = \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - {\sqrt{14}}^{2}}$

$y = - \sqrt{14}$

$x = \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - {\sqrt{14}}^{2}}$

$y = - \sqrt{14}$

Rewrite ${\sqrt{14}}^{2}$ as $14$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{14}$ as $14^{\frac{1}{2}}$ .

$x = \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - {(14^{\frac{1}{2}})}^{2}}$

$y = - \sqrt{14}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - 14^{\frac{1}{2} \cdot 2}}$

$y = - \sqrt{14}$

Combine $\frac{1}{2}$ and $2$ .

$x = \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - 14^{\frac{2}{2}}}$

$y = - \sqrt{14}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - 14^{\frac{2}{2}}}$

$y = - \sqrt{14}$

Rewrite the expression.

$x = \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - 14}$

$y = - \sqrt{14}$

$x = \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - 14}$

$y = - \sqrt{14}$

Evaluate the exponent.

$x = \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - 1 \cdot 14}$

$y = - \sqrt{14}$

$x = \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - 1 \cdot 14}$

$y = - \sqrt{14}$

Multiply $- 1$ by $14$ .

$x = \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - 14}$

$y = - \sqrt{14}$

$x = \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - 14}$

$y = - \sqrt{14}$

Subtract $14$ from $16$ .

$x = \sqrt{2 + 4 \sqrt{14} - 4 \sqrt{14}}$

$y = - \sqrt{14}$

Subtract $4 \sqrt{14}$ from $4 \sqrt{14}$ .

$x = \sqrt{2 + 0}$

$y = - \sqrt{14}$

Add $2$ and $0$ .

$x = \sqrt{2}$

$y = - \sqrt{14}$

$x = \sqrt{2}$

$y = - \sqrt{14}$

$x = \sqrt{2}$

$y = - \sqrt{14}$

$x = \sqrt{2}$

$y = - \sqrt{14}$

$x = \sqrt{2}$

$y = - \sqrt{14}$

$x = \sqrt{2}$

$y = - \sqrt{14}$

$x = \sqrt{2}$

$y = - \sqrt{14}$

Step 3

Solve the system $x = - \sqrt{(4 + y) (4 - y)}, y^{2} - 2 x^{2} = 10$ .

Replace all occurrences of $x$ with $- \sqrt{(4 + y) (4 - y)}$ in each equation.

Replace all occurrences of $x$ in $y^{2} - 2 x^{2} = 10$ with $- \sqrt{(4 + y) (4 - y)}$ .

$y^{2} - 2 {(- \sqrt{(4 + y) (4 - y)})}^{2} = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Simplify the left side.

Simplify $y^{2} - 2 {(- \sqrt{(4 + y) (4 - y)})}^{2}$ .

Simplify each term.

Apply the product rule to $- \sqrt{(4 + y) (4 - y)}$ .

$y^{2} - 2 ({(- 1)}^{2} {\sqrt{(4 + y) (4 - y)}}^{2}) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Raise $- 1$ to the power of $2$ .

$y^{2} - 2 (1 {\sqrt{(4 + y) (4 - y)}}^{2}) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Multiply ${\sqrt{(4 + y) (4 - y)}}^{2}$ by $1$ .

$y^{2} - 2 {\sqrt{(4 + y) (4 - y)}}^{2} = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Rewrite ${\sqrt{(4 + y) (4 - y)}}^{2}$ as $(4 + y) (4 - y)$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(4 + y) (4 - y)}$ as ${((4 + y) (4 - y))}^{\frac{1}{2}}$ .

$y^{2} - 2 {({((4 + y) (4 - y))}^{\frac{1}{2}})}^{2} = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y^{2} - 2 {((4 + y) (4 - y))}^{\frac{1}{2} \cdot 2} = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Combine $\frac{1}{2}$ and $2$ .

$y^{2} - 2 {((4 + y) (4 - y))}^{\frac{2}{2}} = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y^{2} - 2 {((4 + y) (4 - y))}^{\frac{2}{2}} = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Rewrite the expression.

$y^{2} - 2 ((4 + y) (4 - y)) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

$y^{2} - 2 ((4 + y) (4 - y)) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Simplify.

$y^{2} - 2 ((4 + y) (4 - y)) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

$y^{2} - 2 ((4 + y) (4 - y)) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Expand $(4 + y) (4 - y)$ using the FOIL Method.

Apply the distributive property.

$y^{2} - 2 (4 (4 - y) + y (4 - y)) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Apply the distributive property.

$y^{2} - 2 (4 \cdot 4 + 4 (- y) + y (4 - y)) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Apply the distributive property.

$y^{2} - 2 (4 \cdot 4 + 4 (- y) + y \cdot 4 + y (- y)) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

$y^{2} - 2 (4 \cdot 4 + 4 (- y) + y \cdot 4 + y (- y)) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

$y^{2} - 2 (16 + 4 (- y) + y \cdot 4 + y (- y)) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Multiply $- 1$ by $4$ .

$y^{2} - 2 (16 - 4 y + y \cdot 4 + y (- y)) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Move $4$ to the left of $y$ .

$y^{2} - 2 (16 - 4 y + 4 \cdot y + y (- y)) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Rewrite using the commutative property of multiplication.

$y^{2} - 2 (16 - 4 y + 4 y - y \cdot y) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

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

Move $y$ .

$y^{2} - 2 (16 - 4 y + 4 y - (y \cdot y)) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Multiply $y$ by $y$ .

$y^{2} - 2 (16 - 4 y + 4 y - y^{2}) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

$y^{2} - 2 (16 - 4 y + 4 y - y^{2}) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

$y^{2} - 2 (16 - 4 y + 4 y - y^{2}) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Add $- 4 y$ and $4 y$ .

$y^{2} - 2 (16 + 0 - y^{2}) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Add $16$ and $0$ .

$y^{2} - 2 (16 - y^{2}) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

$y^{2} - 2 (16 - y^{2}) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Apply the distributive property.

$y^{2} - 2 \cdot 16 - 2 (- y^{2}) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Multiply $- 2$ by $16$ .

$y^{2} - 32 - 2 (- y^{2}) = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Multiply $- 1$ by $- 2$ .

$y^{2} - 32 + 2 y^{2} = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

$y^{2} - 32 + 2 y^{2} = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

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

$3 y^{2} - 32 = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

$3 y^{2} - 32 = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

$3 y^{2} - 32 = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

$3 y^{2} - 32 = 10$

$x = - \sqrt{(4 + y) (4 - y)}$

Solve for $y$ in $3 y^{2} - 32 = 10$ .

Move all terms not containing $y$ to the right side of the equation.

Add $32$ to both sides of the equation.

$3 y^{2} = 10 + 32$

$x = - \sqrt{(4 + y) (4 - y)}$

Add $10$ and $32$ .

$3 y^{2} = 42$

$x = - \sqrt{(4 + y) (4 - y)}$

$3 y^{2} = 42$

$x = - \sqrt{(4 + y) (4 - y)}$

Divide each term in $3 y^{2} = 42$ by $3$ and simplify.

Divide each term in $3 y^{2} = 42$ by $3$ .

$\frac{3 y^{2}}{3} = \frac{42}{3}$

$x = - \sqrt{(4 + y) (4 - y)}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y^{2}}{3} = \frac{42}{3}$

$x = - \sqrt{(4 + y) (4 - y)}$

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{42}{3}$

$x = - \sqrt{(4 + y) (4 - y)}$

$y^{2} = \frac{42}{3}$

$x = - \sqrt{(4 + y) (4 - y)}$

$y^{2} = \frac{42}{3}$

$x = - \sqrt{(4 + y) (4 - y)}$

Simplify the right side.

Divide $42$ by $3$ .

$y^{2} = 14$

$x = - \sqrt{(4 + y) (4 - y)}$

$y^{2} = 14$

$x = - \sqrt{(4 + y) (4 - y)}$

$y^{2} = 14$

$x = - \sqrt{(4 + y) (4 - y)}$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{14}$

$x = - \sqrt{(4 + y) (4 - y)}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{14}$

$x = - \sqrt{(4 + y) (4 - y)}$

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{14}$

$x = - \sqrt{(4 + y) (4 - y)}$

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{14}, - \sqrt{14}$

$x = - \sqrt{(4 + y) (4 - y)}$

$y = \sqrt{14}, - \sqrt{14}$

$x = - \sqrt{(4 + y) (4 - y)}$

$y = \sqrt{14}, - \sqrt{14}$

$x = - \sqrt{(4 + y) (4 - y)}$

Replace all occurrences of $y$ with $\sqrt{14}$ in each equation.

Replace all occurrences of $y$ in $x = - \sqrt{(4 + y) (4 - y)}$ with $\sqrt{14}$ .

$x = - \sqrt{(4 + \sqrt{14}) (4 - (\sqrt{14}))}$

$y = \sqrt{14}$

Simplify $x = - \sqrt{(4 + \sqrt{14}) (4 - (\sqrt{14}))}$ .

Simplify the left side.

Remove parentheses.

$x = - \sqrt{(4 + \sqrt{14}) (4 - (\sqrt{14}))}$

$y = \sqrt{14}$

$x = - \sqrt{(4 + \sqrt{14}) (4 - (\sqrt{14}))}$

$y = \sqrt{14}$

Simplify the right side.

Simplify $- \sqrt{(4 + \sqrt{14}) (4 - (\sqrt{14}))}$ .

Expand $(4 + \sqrt{14}) (4 - \sqrt{14})$ using the FOIL Method.

Apply the distributive property.

$x = - \sqrt{4 (4 - \sqrt{14}) + \sqrt{14} (4 - \sqrt{14})}$

$y = \sqrt{14}$

Apply the distributive property.

$x = - \sqrt{4 \cdot 4 + 4 (- \sqrt{14}) + \sqrt{14} (4 - \sqrt{14})}$

$y = \sqrt{14}$

Apply the distributive property.

$x = - \sqrt{4 \cdot 4 + 4 (- \sqrt{14}) + \sqrt{14} \cdot 4 + \sqrt{14} (- \sqrt{14})}$

$y = \sqrt{14}$

$x = - \sqrt{4 \cdot 4 + 4 (- \sqrt{14}) + \sqrt{14} \cdot 4 + \sqrt{14} (- \sqrt{14})}$

$y = \sqrt{14}$

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

$x = - \sqrt{16 + 4 (- \sqrt{14}) + \sqrt{14} \cdot 4 + \sqrt{14} (- \sqrt{14})}$

$y = \sqrt{14}$

Multiply $- 1$ by $4$ .

$x = - \sqrt{16 - 4 \sqrt{14} + \sqrt{14} \cdot 4 + \sqrt{14} (- \sqrt{14})}$

$y = \sqrt{14}$

Move $4$ to the left of $\sqrt{14}$ .

$x = - \sqrt{16 - 4 \sqrt{14} + 4 \cdot \sqrt{14} + \sqrt{14} (- \sqrt{14})}$

$y = \sqrt{14}$

Multiply $\sqrt{14} (- \sqrt{14})$ .

Raise $\sqrt{14}$ to the power of $1$ .

$x = - \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - (\sqrt{14} \sqrt{14})}$

$y = \sqrt{14}$

Raise $\sqrt{14}$ to the power of $1$ .

$x = - \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - (\sqrt{14} \sqrt{14})}$

$y = \sqrt{14}$

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

$x = - \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - {\sqrt{14}}^{1 + 1}}$

$y = \sqrt{14}$

Add $1$ and $1$ .

$x = - \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - {\sqrt{14}}^{2}}$

$y = \sqrt{14}$

$x = - \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - {\sqrt{14}}^{2}}$

$y = \sqrt{14}$

Rewrite ${\sqrt{14}}^{2}$ as $14$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{14}$ as $14^{\frac{1}{2}}$ .

$x = - \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - {(14^{\frac{1}{2}})}^{2}}$

$y = \sqrt{14}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = - \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - 14^{\frac{1}{2} \cdot 2}}$

$y = \sqrt{14}$

Combine $\frac{1}{2}$ and $2$ .

$x = - \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - 14^{\frac{2}{2}}}$

$y = \sqrt{14}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = - \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - 14^{\frac{2}{2}}}$

$y = \sqrt{14}$

Rewrite the expression.

$x = - \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - 14}$

$y = \sqrt{14}$

$x = - \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - 14}$

$y = \sqrt{14}$

Evaluate the exponent.

$x = - \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - 1 \cdot 14}$

$y = \sqrt{14}$

$x = - \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - 1 \cdot 14}$

$y = \sqrt{14}$

Multiply $- 1$ by $14$ .

$x = - \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - 14}$

$y = \sqrt{14}$

$x = - \sqrt{16 - 4 \sqrt{14} + 4 \sqrt{14} - 14}$

$y = \sqrt{14}$

Subtract $14$ from $16$ .

$x = - \sqrt{2 - 4 \sqrt{14} + 4 \sqrt{14}}$

$y = \sqrt{14}$

Add $- 4 \sqrt{14}$ and $4 \sqrt{14}$ .

$x = - \sqrt{2 + 0}$

$y = \sqrt{14}$

Add $2$ and $0$ .

$x = - \sqrt{2}$

$y = \sqrt{14}$

$x = - \sqrt{2}$

$y = \sqrt{14}$

$x = - \sqrt{2}$

$y = \sqrt{14}$

$x = - \sqrt{2}$

$y = \sqrt{14}$

$x = - \sqrt{2}$

$y = \sqrt{14}$

$x = - \sqrt{2}$

$y = \sqrt{14}$

Replace all occurrences of $y$ with $- \sqrt{14}$ in each equation.

Replace all occurrences of $y$ in $x = - \sqrt{(4 + y) (4 - y)}$ with $- \sqrt{14}$ .

$x = - \sqrt{(4 - \sqrt{14}) (4 - (- \sqrt{14}))}$

$y = - \sqrt{14}$

Simplify $x = - \sqrt{(4 - \sqrt{14}) (4 - (- \sqrt{14}))}$ .

Simplify the left side.

Remove parentheses.

$x = - \sqrt{(4 - \sqrt{14}) (4 - (- \sqrt{14}))}$

$y = - \sqrt{14}$

$x = - \sqrt{(4 - \sqrt{14}) (4 - (- \sqrt{14}))}$

$y = - \sqrt{14}$

Simplify the right side.

Simplify $- \sqrt{(4 - \sqrt{14}) (4 - (- \sqrt{14}))}$ .

Multiply $- - \sqrt{14}$ .

Multiply $- 1$ by $- 1$ .

$x = - \sqrt{(4 - \sqrt{14}) (4 + 1 \sqrt{14})}$

$y = - \sqrt{14}$

Multiply $\sqrt{14}$ by $1$ .

$x = - \sqrt{(4 - \sqrt{14}) (4 + \sqrt{14})}$

$y = - \sqrt{14}$

$x = - \sqrt{(4 - \sqrt{14}) (4 + \sqrt{14})}$

$y = - \sqrt{14}$

Expand $(4 - \sqrt{14}) (4 + \sqrt{14})$ using the FOIL Method.

Apply the distributive property.

$x = - \sqrt{4 (4 + \sqrt{14}) - \sqrt{14} (4 + \sqrt{14})}$

$y = - \sqrt{14}$

Apply the distributive property.

$x = - \sqrt{4 \cdot 4 + 4 \sqrt{14} - \sqrt{14} (4 + \sqrt{14})}$

$y = - \sqrt{14}$

Apply the distributive property.

$x = - \sqrt{4 \cdot 4 + 4 \sqrt{14} - \sqrt{14} \cdot 4 - \sqrt{14} \sqrt{14}}$

$y = - \sqrt{14}$

$x = - \sqrt{4 \cdot 4 + 4 \sqrt{14} - \sqrt{14} \cdot 4 - \sqrt{14} \sqrt{14}}$

$y = - \sqrt{14}$

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

$x = - \sqrt{16 + 4 \sqrt{14} - \sqrt{14} \cdot 4 - \sqrt{14} \sqrt{14}}$

$y = - \sqrt{14}$

Multiply $4$ by $- 1$ .

$x = - \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - \sqrt{14} \sqrt{14}}$

$y = - \sqrt{14}$

Multiply $- \sqrt{14} \sqrt{14}$ .

Raise $\sqrt{14}$ to the power of $1$ .

$x = - \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - (\sqrt{14} \sqrt{14})}$

$y = - \sqrt{14}$

Raise $\sqrt{14}$ to the power of $1$ .

$x = - \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - (\sqrt{14} \sqrt{14})}$

$y = - \sqrt{14}$

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

$x = - \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - {\sqrt{14}}^{1 + 1}}$

$y = - \sqrt{14}$

Add $1$ and $1$ .

$x = - \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - {\sqrt{14}}^{2}}$

$y = - \sqrt{14}$

$x = - \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - {\sqrt{14}}^{2}}$

$y = - \sqrt{14}$

Rewrite ${\sqrt{14}}^{2}$ as $14$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{14}$ as $14^{\frac{1}{2}}$ .

$x = - \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - {(14^{\frac{1}{2}})}^{2}}$

$y = - \sqrt{14}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = - \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - 14^{\frac{1}{2} \cdot 2}}$

$y = - \sqrt{14}$

Combine $\frac{1}{2}$ and $2$ .

$x = - \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - 14^{\frac{2}{2}}}$

$y = - \sqrt{14}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = - \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - 14^{\frac{2}{2}}}$

$y = - \sqrt{14}$

Rewrite the expression.

$x = - \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - 14}$

$y = - \sqrt{14}$

$x = - \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - 14}$

$y = - \sqrt{14}$

Evaluate the exponent.

$x = - \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - 1 \cdot 14}$

$y = - \sqrt{14}$

$x = - \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - 1 \cdot 14}$

$y = - \sqrt{14}$

Multiply $- 1$ by $14$ .

$x = - \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - 14}$

$y = - \sqrt{14}$

$x = - \sqrt{16 + 4 \sqrt{14} - 4 \sqrt{14} - 14}$

$y = - \sqrt{14}$

Subtract $14$ from $16$ .

$x = - \sqrt{2 + 4 \sqrt{14} - 4 \sqrt{14}}$

$y = - \sqrt{14}$

Subtract $4 \sqrt{14}$ from $4 \sqrt{14}$ .

$x = - \sqrt{2 + 0}$

$y = - \sqrt{14}$

Add $2$ and $0$ .

$x = - \sqrt{2}$

$y = - \sqrt{14}$

$x = - \sqrt{2}$

$y = - \sqrt{14}$

$x = - \sqrt{2}$

$y = - \sqrt{14}$

$x = - \sqrt{2}$

$y = - \sqrt{14}$

$x = - \sqrt{2}$

$y = - \sqrt{14}$

$x = - \sqrt{2}$

$y = - \sqrt{14}$

$x = - \sqrt{2}$

$y = - \sqrt{14}$

Step 4

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

$(\sqrt{2}, \sqrt{14})$

$(\sqrt{2}, - \sqrt{14})$

$(- \sqrt{2}, \sqrt{14})$

$(- \sqrt{2}, - \sqrt{14})$

Step 5

The result can be shown in multiple forms.

Point Form:

$(\sqrt{2}, \sqrt{14}), (\sqrt{2}, - \sqrt{14}), (- \sqrt{2}, \sqrt{14}), (- \sqrt{2}, - \sqrt{14})$

Equation Form:

$x = \sqrt{2}, y = \sqrt{14}$

$x = \sqrt{2}, y = - \sqrt{14}$

$x = - \sqrt{2}, y = \sqrt{14}$

$x = - \sqrt{2}, y = - \sqrt{14}$

Step 6

Solve by Substitution x^2+y^2=16 , y^2-2x^2=10

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