Solve by Substitution 8x^2+y^2=72 , xy=8

$8 x^{2} + y^{2} = 72$ , $x y = 8$

Step 1

Divide each term in $x y = 8$ by $y$ and simplify.

Divide each term in $x y = 8$ by $y$ .

$\frac{x y}{y} = \frac{8}{y}$

$8 x^{2} + y^{2} = 72$

Simplify the left side.

Cancel the common factor of $y$ .

Cancel the common factor.

$\frac{x y}{y} = \frac{8}{y}$

$8 x^{2} + y^{2} = 72$

Divide $x$ by $1$ .

$x = \frac{8}{y}$

$8 x^{2} + y^{2} = 72$

$x = \frac{8}{y}$

$8 x^{2} + y^{2} = 72$

$x = \frac{8}{y}$

$8 x^{2} + y^{2} = 72$

$x = \frac{8}{y}$

$8 x^{2} + y^{2} = 72$

Step 2

Replace all occurrences of $x$ with $\frac{8}{y}$ in each equation.

Replace all occurrences of $x$ in $8 x^{2} + y^{2} = 72$ with $\frac{8}{y}$ .

$8 {(\frac{8}{y})}^{2} + y^{2} = 72$

$x = \frac{8}{y}$

Simplify the left side.

Simplify each term.

Apply the product rule to $\frac{8}{y}$ .

$8 (\frac{8^{2}}{y^{2}}) + y^{2} = 72$

$x = \frac{8}{y}$

Raise $8$ to the power of $2$ .

$8 (\frac{64}{y^{2}}) + y^{2} = 72$

$x = \frac{8}{y}$

Multiply $8 \frac{64}{y^{2}}$ .

Combine $8$ and $\frac{64}{y^{2}}$ .

$\frac{8 \cdot 64}{y^{2}} + y^{2} = 72$

$x = \frac{8}{y}$

Multiply $8$ by $64$ .

$\frac{512}{y^{2}} + y^{2} = 72$

$x = \frac{8}{y}$

$\frac{512}{y^{2}} + y^{2} = 72$

$x = \frac{8}{y}$

$\frac{512}{y^{2}} + y^{2} = 72$

$x = \frac{8}{y}$

$\frac{512}{y^{2}} + y^{2} = 72$

$x = \frac{8}{y}$

$\frac{512}{y^{2}} + y^{2} = 72$

$x = \frac{8}{y}$

Step 3

Solve for $y$ in $\frac{512}{y^{2}} + y^{2} = 72$ .

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y^{2}, 1, 1$

$x = \frac{8}{y}$

The LCM of one and any expression is the expression.

$y^{2}$

$x = \frac{8}{y}$

$y^{2}$

$x = \frac{8}{y}$

Multiply each term in $\frac{512}{y^{2}} + y^{2} = 72$ by $y^{2}$ to eliminate the fractions.

Multiply each term in $\frac{512}{y^{2}} + y^{2} = 72$ by $y^{2}$ .

$\frac{512}{y^{2}} \cdot y^{2} + y^{2} y^{2} = 72 y^{2}$

$x = \frac{8}{y}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $y^{2}$ .

Cancel the common factor.

$\frac{512}{y^{2}} \cdot y^{2} + y^{2} y^{2} = 72 y^{2}$

$x = \frac{8}{y}$

Rewrite the expression.

$512 + y^{2} y^{2} = 72 y^{2}$

$x = \frac{8}{y}$

$512 + y^{2} y^{2} = 72 y^{2}$

$x = \frac{8}{y}$

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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

$512 + y^{2 + 2} = 72 y^{2}$

$x = \frac{8}{y}$

Add $2$ and $2$ .

$512 + y^{4} = 72 y^{2}$

$x = \frac{8}{y}$

$512 + y^{4} = 72 y^{2}$

$x = \frac{8}{y}$

$512 + y^{4} = 72 y^{2}$

$x = \frac{8}{y}$

$512 + y^{4} = 72 y^{2}$

$x = \frac{8}{y}$

$512 + y^{4} = 72 y^{2}$

$x = \frac{8}{y}$

Solve the equation.

Subtract $72 y^{2}$ from both sides of the equation.

$512 + y^{4} - 72 y^{2} = 0$

$x = \frac{8}{y}$

Substitute $u = y^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} - 72 u + 512 = 0$

$u = y^{2}$

$x = \frac{8}{y}$

Factor $u^{2} - 72 u + 512$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $512$ and whose sum is $- 72$ .

$- 64, - 8$

$x = \frac{8}{y}$

Write the factored form using these integers.

$(u - 64) (u - 8) = 0$

$x = \frac{8}{y}$

$(u - 64) (u - 8) = 0$

$x = \frac{8}{y}$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$u - 64 = 0$

$u - 8 = 0$

$x = \frac{8}{y}$

Set $u - 64$ equal to $0$ and solve for $u$ .

Set $u - 64$ equal to $0$ .

$u - 64 = 0$

$x = \frac{8}{y}$

Add $64$ to both sides of the equation.

$u = 64$

$x = \frac{8}{y}$

$u = 64$

$x = \frac{8}{y}$

Set $u - 8$ equal to $0$ and solve for $u$ .

Set $u - 8$ equal to $0$ .

$u - 8 = 0$

$x = \frac{8}{y}$

Add $8$ to both sides of the equation.

$u = 8$

$x = \frac{8}{y}$

$u = 8$

$x = \frac{8}{y}$

The final solution is all the values that make $(u - 64) (u - 8) = 0$ true.

$u = 64, 8$

$x = \frac{8}{y}$

Substitute the real value of $u = y^{2}$ back into the solved equation.

$y^{2} = 64$

$y^{2} = 8$

$x = \frac{8}{y}$

Solve the first equation for $y$ .

$y^{2} = 64$

$x = \frac{8}{y}$

Solve the equation for $y$ .

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

$y = \pm \sqrt{64}$

$x = \frac{8}{y}$

Simplify $\pm \sqrt{64}$ .

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

$y = \pm \sqrt{8^{2}}$

$x = \frac{8}{y}$

Pull terms out from under the radical, assuming positive real numbers.

$y = \pm 8$

$x = \frac{8}{y}$

$y = \pm 8$

$x = \frac{8}{y}$

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.

$y = 8$

$x = \frac{8}{y}$

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

$y = - 8$

$x = \frac{8}{y}$

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

$y = 8, - 8$

$x = \frac{8}{y}$

$y = 8, - 8$

$x = \frac{8}{y}$

$y = 8, - 8$

$x = \frac{8}{y}$

Solve the second equation for $y$ .

$y^{2} = 8$

$x = \frac{8}{y}$

Solve the equation for $y$ .

Remove parentheses.

$y^{2} = 8$

$x = \frac{8}{y}$

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

$y = \pm \sqrt{8}$

$x = \frac{8}{y}$

Simplify $\pm \sqrt{8}$ .

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

Factor $4$ out of $8$ .

$y = \pm \sqrt{4 (2)}$

$x = \frac{8}{y}$

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

$y = \pm \sqrt{2^{2} \cdot 2}$

$x = \frac{8}{y}$

$y = \pm \sqrt{2^{2} \cdot 2}$

$x = \frac{8}{y}$

Pull terms out from under the radical.

$y = \pm 2 \sqrt{2}$

$x = \frac{8}{y}$

$y = \pm 2 \sqrt{2}$

$x = \frac{8}{y}$

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.

$y = 2 \sqrt{2}$

$x = \frac{8}{y}$

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

$y = - 2 \sqrt{2}$

$x = \frac{8}{y}$

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

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

$x = \frac{8}{y}$

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

$x = \frac{8}{y}$

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

$x = \frac{8}{y}$

The solution to $y^{4} - 72 y^{2} + 512 = 0$ is $y = 8, - 8, 2 \sqrt{2}, - 2 \sqrt{2}$ .

$y = 8, - 8, 2 \sqrt{2}, - 2 \sqrt{2}$

$x = \frac{8}{y}$

$y = 8, - 8, 2 \sqrt{2}, - 2 \sqrt{2}$

$x = \frac{8}{y}$

$y = 8, - 8, 2 \sqrt{2}, - 2 \sqrt{2}$

$x = \frac{8}{y}$

Step 4

Replace all occurrences of $y$ with $8$ in each equation.

Replace all occurrences of $y$ in $x = \frac{8}{y}$ with $8$ .

$x = \frac{8}{8}$

$y = 8$

Simplify the right side.

Divide $8$ by $8$ .

$x = 1$

$y = 8$

$x = 1$

$y = 8$

$x = 1$

$y = 8$

Step 5

Replace all occurrences of $y$ with $- 8$ in each equation.

Replace all occurrences of $y$ in $x = \frac{8}{y}$ with $- 8$ .

$x = \frac{8}{- 8}$

$y = - 8$

Simplify the right side.

Divide $8$ by $- 8$ .

$x = - 1$

$y = - 8$

$x = - 1$

$y = - 8$

$x = - 1$

$y = - 8$

Step 6

Replace all occurrences of $y$ with $8$ in each equation.

Replace all occurrences of $y$ in $x = \frac{8}{y}$ with $8$ .

$x = \frac{8}{8}$

$y = 8$

Simplify the right side.

Divide $8$ by $8$ .

$x = 1$

$y = 8$

$x = 1$

$y = 8$

$x = 1$

$y = 8$

Step 7

Replace all occurrences of $y$ with $- 8$ in each equation.

Replace all occurrences of $y$ in $x = \frac{8}{y}$ with $- 8$ .

$x = \frac{8}{- 8}$

$y = - 8$

Simplify the right side.

Divide $8$ by $- 8$ .

$x = - 1$

$y = - 8$

$x = - 1$

$y = - 8$

$x = - 1$

$y = - 8$

Step 8

Replace all occurrences of $y$ with $2 \sqrt{2}$ in each equation.

Replace all occurrences of $y$ in $x = \frac{8}{y}$ with $2 \sqrt{2}$ .

$x = \frac{8}{2 \sqrt{2}}$

$y = 2 \sqrt{2}$

Simplify the right side.

Simplify $\frac{8}{2 \sqrt{2}}$ .

Cancel the common factor of $8$ and $2$ .

Factor $2$ out of $8$ .

$x = \frac{2 \cdot 4}{2 \sqrt{2}}$

$y = 2 \sqrt{2}$

Cancel the common factors.

Factor $2$ out of $2 \sqrt{2}$ .

$x = \frac{2 \cdot 4}{2 (\sqrt{2})}$

$y = 2 \sqrt{2}$

Cancel the common factor.

$x = \frac{2 \cdot 4}{2 \sqrt{2}}$

$y = 2 \sqrt{2}$

Rewrite the expression.

$x = \frac{4}{\sqrt{2}}$

$y = 2 \sqrt{2}$

$x = \frac{4}{\sqrt{2}}$

$y = 2 \sqrt{2}$

$x = \frac{4}{\sqrt{2}}$

$y = 2 \sqrt{2}$

Multiply $\frac{4}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \frac{4}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

$y = 2 \sqrt{2}$

Combine and simplify the denominator.

Multiply $\frac{4}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \frac{4 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

$y = 2 \sqrt{2}$

Raise $\sqrt{2}$ to the power of $1$ .

$x = \frac{4 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

$y = 2 \sqrt{2}$

Raise $\sqrt{2}$ to the power of $1$ .

$x = \frac{4 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

$y = 2 \sqrt{2}$

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

$x = \frac{4 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

$y = 2 \sqrt{2}$

Add $1$ and $1$ .

$x = \frac{4 \sqrt{2}}{{\sqrt{2}}^{2}}$

$y = 2 \sqrt{2}$

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

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

$x = \frac{4 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

$y = 2 \sqrt{2}$

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

$x = \frac{4 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

$y = 2 \sqrt{2}$

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

$x = \frac{4 \sqrt{2}}{2^{\frac{2}{2}}}$

$y = 2 \sqrt{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \frac{4 \sqrt{2}}{2^{\frac{2}{2}}}$

$y = 2 \sqrt{2}$

Rewrite the expression.

$x = \frac{4 \sqrt{2}}{2}$

$y = 2 \sqrt{2}$

$x = \frac{4 \sqrt{2}}{2}$

$y = 2 \sqrt{2}$

Evaluate the exponent.

$x = \frac{4 \sqrt{2}}{2}$

$y = 2 \sqrt{2}$

$x = \frac{4 \sqrt{2}}{2}$

$y = 2 \sqrt{2}$

$x = \frac{4 \sqrt{2}}{2}$

$y = 2 \sqrt{2}$

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4 \sqrt{2}$ .

$x = \frac{2 (2 \sqrt{2})}{2}$

$y = 2 \sqrt{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$x = \frac{2 (2 \sqrt{2})}{2 (1)}$

$y = 2 \sqrt{2}$

Cancel the common factor.

$x = \frac{2 (2 \sqrt{2})}{2 \cdot 1}$

$y = 2 \sqrt{2}$

Rewrite the expression.

$x = \frac{2 \sqrt{2}}{1}$

$y = 2 \sqrt{2}$

Divide $2 \sqrt{2}$ by $1$ .

$x = 2 \sqrt{2}$

$y = 2 \sqrt{2}$

$x = 2 \sqrt{2}$

$y = 2 \sqrt{2}$

$x = 2 \sqrt{2}$

$y = 2 \sqrt{2}$

$x = 2 \sqrt{2}$

$y = 2 \sqrt{2}$

$x = 2 \sqrt{2}$

$y = 2 \sqrt{2}$

$x = 2 \sqrt{2}$

$y = 2 \sqrt{2}$

Step 9

Replace all occurrences of $y$ with $8$ in each equation.

Replace all occurrences of $y$ in $x = \frac{8}{y}$ with $8$ .

$x = \frac{8}{8}$

$y = 8$

Simplify the right side.

Divide $8$ by $8$ .

$x = 1$

$y = 8$

$x = 1$

$y = 8$

$x = 1$

$y = 8$

Step 10

Replace all occurrences of $y$ with $- 8$ in each equation.

Replace all occurrences of $y$ in $x = \frac{8}{y}$ with $- 8$ .

$x = \frac{8}{- 8}$

$y = - 8$

Simplify the right side.

Divide $8$ by $- 8$ .

$x = - 1$

$y = - 8$

$x = - 1$

$y = - 8$

$x = - 1$

$y = - 8$

Step 11

Replace all occurrences of $y$ with $2 \sqrt{2}$ in each equation.

Replace all occurrences of $y$ in $x = \frac{8}{y}$ with $2 \sqrt{2}$ .

$x = \frac{8}{2 \sqrt{2}}$

$y = 2 \sqrt{2}$

Simplify the right side.

Simplify $\frac{8}{2 \sqrt{2}}$ .

Cancel the common factor of $8$ and $2$ .

Factor $2$ out of $8$ .

$x = \frac{2 \cdot 4}{2 \sqrt{2}}$

$y = 2 \sqrt{2}$

Cancel the common factors.

Factor $2$ out of $2 \sqrt{2}$ .

$x = \frac{2 \cdot 4}{2 (\sqrt{2})}$

$y = 2 \sqrt{2}$

Cancel the common factor.

$x = \frac{2 \cdot 4}{2 \sqrt{2}}$

$y = 2 \sqrt{2}$

Rewrite the expression.

$x = \frac{4}{\sqrt{2}}$

$y = 2 \sqrt{2}$

$x = \frac{4}{\sqrt{2}}$

$y = 2 \sqrt{2}$

$x = \frac{4}{\sqrt{2}}$

$y = 2 \sqrt{2}$

Multiply $\frac{4}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \frac{4}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

$y = 2 \sqrt{2}$

Combine and simplify the denominator.

Multiply $\frac{4}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \frac{4 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

$y = 2 \sqrt{2}$

Raise $\sqrt{2}$ to the power of $1$ .

$x = \frac{4 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

$y = 2 \sqrt{2}$

Raise $\sqrt{2}$ to the power of $1$ .

$x = \frac{4 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

$y = 2 \sqrt{2}$

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

$x = \frac{4 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

$y = 2 \sqrt{2}$

Add $1$ and $1$ .

$x = \frac{4 \sqrt{2}}{{\sqrt{2}}^{2}}$

$y = 2 \sqrt{2}$

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

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

$x = \frac{4 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

$y = 2 \sqrt{2}$

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

$x = \frac{4 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

$y = 2 \sqrt{2}$

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

$x = \frac{4 \sqrt{2}}{2^{\frac{2}{2}}}$

$y = 2 \sqrt{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \frac{4 \sqrt{2}}{2^{\frac{2}{2}}}$

$y = 2 \sqrt{2}$

Rewrite the expression.

$x = \frac{4 \sqrt{2}}{2}$

$y = 2 \sqrt{2}$

$x = \frac{4 \sqrt{2}}{2}$

$y = 2 \sqrt{2}$

Evaluate the exponent.

$x = \frac{4 \sqrt{2}}{2}$

$y = 2 \sqrt{2}$

$x = \frac{4 \sqrt{2}}{2}$

$y = 2 \sqrt{2}$

$x = \frac{4 \sqrt{2}}{2}$

$y = 2 \sqrt{2}$

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4 \sqrt{2}$ .

$x = \frac{2 (2 \sqrt{2})}{2}$

$y = 2 \sqrt{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$x = \frac{2 (2 \sqrt{2})}{2 (1)}$

$y = 2 \sqrt{2}$

Cancel the common factor.

$x = \frac{2 (2 \sqrt{2})}{2 \cdot 1}$

$y = 2 \sqrt{2}$

Rewrite the expression.

$x = \frac{2 \sqrt{2}}{1}$

$y = 2 \sqrt{2}$

Divide $2 \sqrt{2}$ by $1$ .

$x = 2 \sqrt{2}$

$y = 2 \sqrt{2}$

$x = 2 \sqrt{2}$

$y = 2 \sqrt{2}$

$x = 2 \sqrt{2}$

$y = 2 \sqrt{2}$

$x = 2 \sqrt{2}$

$y = 2 \sqrt{2}$

$x = 2 \sqrt{2}$

$y = 2 \sqrt{2}$

$x = 2 \sqrt{2}$

$y = 2 \sqrt{2}$

Step 12

Replace all occurrences of $y$ with $- 2 \sqrt{2}$ in each equation.

Replace all occurrences of $y$ in $x = \frac{8}{y}$ with $- 2 \sqrt{2}$ .

$x = \frac{8}{- 2 \sqrt{2}}$

$y = - 2 \sqrt{2}$

Simplify the right side.

Simplify $\frac{8}{- 2 \sqrt{2}}$ .

Cancel the common factor of $8$ and $- 2$ .

Factor $2$ out of $8$ .

$x = \frac{2 (4)}{- 2 \sqrt{2}}$

$y = - 2 \sqrt{2}$

Cancel the common factors.

Factor $2$ out of $- 2 \sqrt{2}$ .

$x = \frac{2 (4)}{2 (- \sqrt{2})}$

$y = - 2 \sqrt{2}$

Cancel the common factor.

$x = \frac{2 \cdot 4}{2 (- \sqrt{2})}$

$y = - 2 \sqrt{2}$

Rewrite the expression.

$x = \frac{4}{- \sqrt{2}}$

$y = - 2 \sqrt{2}$

$x = \frac{4}{- \sqrt{2}}$

$y = - 2 \sqrt{2}$

$x = \frac{4}{- \sqrt{2}}$

$y = - 2 \sqrt{2}$

Move the negative in front of the fraction.

$x = - \frac{4}{\sqrt{2}}$

$y = - 2 \sqrt{2}$

Multiply $\frac{4}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = - (\frac{4}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

$y = - 2 \sqrt{2}$

Combine and simplify the denominator.

Multiply $\frac{4}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = - \frac{4 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

$y = - 2 \sqrt{2}$

Raise $\sqrt{2}$ to the power of $1$ .

$x = - \frac{4 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

$y = - 2 \sqrt{2}$

Raise $\sqrt{2}$ to the power of $1$ .

$x = - \frac{4 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

$y = - 2 \sqrt{2}$

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

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

$y = - 2 \sqrt{2}$

Add $1$ and $1$ .

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

$y = - 2 \sqrt{2}$

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

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

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

$y = - 2 \sqrt{2}$

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

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

$y = - 2 \sqrt{2}$

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

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

$y = - 2 \sqrt{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

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

$y = - 2 \sqrt{2}$

Rewrite the expression.

$x = - \frac{4 \sqrt{2}}{2}$

$y = - 2 \sqrt{2}$

$x = - \frac{4 \sqrt{2}}{2}$

$y = - 2 \sqrt{2}$

Evaluate the exponent.

$x = - \frac{4 \sqrt{2}}{2}$

$y = - 2 \sqrt{2}$

$x = - \frac{4 \sqrt{2}}{2}$

$y = - 2 \sqrt{2}$

$x = - \frac{4 \sqrt{2}}{2}$

$y = - 2 \sqrt{2}$

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4 \sqrt{2}$ .

$x = - \frac{2 (2 \sqrt{2})}{2}$

$y = - 2 \sqrt{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$x = - \frac{2 (2 \sqrt{2})}{2 (1)}$

$y = - 2 \sqrt{2}$

Cancel the common factor.

$x = - \frac{2 (2 \sqrt{2})}{2 \cdot 1}$

$y = - 2 \sqrt{2}$

Rewrite the expression.

$x = - \frac{2 \sqrt{2}}{1}$

$y = - 2 \sqrt{2}$

Divide $2 \sqrt{2}$ by $1$ .

$x = - (2 \sqrt{2})$

$y = - 2 \sqrt{2}$

$x = - (2 \sqrt{2})$

$y = - 2 \sqrt{2}$

$x = - (2 \sqrt{2})$

$y = - 2 \sqrt{2}$

Multiply $2$ by $- 1$ .

$x = - 2 \sqrt{2}$

$y = - 2 \sqrt{2}$

$x = - 2 \sqrt{2}$

$y = - 2 \sqrt{2}$

$x = - 2 \sqrt{2}$

$y = - 2 \sqrt{2}$

$x = - 2 \sqrt{2}$

$y = - 2 \sqrt{2}$

Step 13

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

$(1, 8)$

$(- 1, - 8)$

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

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

Step 14

The result can be shown in multiple forms.

Point Form:

$(1, 8), (- 1, - 8), (2 \sqrt{2}, 2 \sqrt{2}), (- 2 \sqrt{2}, - 2 \sqrt{2})$

Equation Form:

$x = 1, y = 8$

$x = - 1, y = - 8$

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

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

Step 15

Solve by Substitution 8x^2+y^2=72 , xy=8

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