Solve by Substitution x^2+y^2=36 , y^2-5x=36

$x^{2} + y^{2} = 36$ , $y^{2} - 5 x = 36$

Step 1

Solve for $x$ in $y^{2} - 5 x = 36$ .

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

$- 5 x = 36 - y^{2}$

$x^{2} + y^{2} = 36$

Divide each term in $- 5 x = 36 - y^{2}$ by $- 5$ and simplify.

Divide each term in $- 5 x = 36 - y^{2}$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{36}{- 5} + \frac{- y^{2}}{- 5}$

$x^{2} + y^{2} = 36$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{36}{- 5} + \frac{- y^{2}}{- 5}$

$x^{2} + y^{2} = 36$

Divide $x$ by $1$ .

$x = \frac{36}{- 5} + \frac{- y^{2}}{- 5}$

$x^{2} + y^{2} = 36$

$x = \frac{36}{- 5} + \frac{- y^{2}}{- 5}$

$x^{2} + y^{2} = 36$

$x = \frac{36}{- 5} + \frac{- y^{2}}{- 5}$

$x^{2} + y^{2} = 36$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{36}{5} + \frac{- y^{2}}{- 5}$

$x^{2} + y^{2} = 36$

Dividing two negative values results in a positive value.

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$x^{2} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$x^{2} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$x^{2} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$x^{2} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$x^{2} + y^{2} = 36$

Step 2

Replace all occurrences of $x$ with $- \frac{36}{5} + \frac{y^{2}}{5}$ in each equation.

Replace all occurrences of $x$ in $x^{2} + y^{2} = 36$ with $- \frac{36}{5} + \frac{y^{2}}{5}$ .

${(- \frac{36}{5} + \frac{y^{2}}{5})}^{2} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Simplify the left side.

Simplify ${(- \frac{36}{5} + \frac{y^{2}}{5})}^{2} + y^{2}$ .

Simplify each term.

Rewrite ${(- \frac{36}{5} + \frac{y^{2}}{5})}^{2}$ as $(- \frac{36}{5} + \frac{y^{2}}{5}) (- \frac{36}{5} + \frac{y^{2}}{5})$ .

$(- \frac{36}{5} + \frac{y^{2}}{5}) (- \frac{36}{5} + \frac{y^{2}}{5}) + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Expand $(- \frac{36}{5} + \frac{y^{2}}{5}) (- \frac{36}{5} + \frac{y^{2}}{5})$ using the FOIL Method.

Apply the distributive property.

$- \frac{36}{5} \cdot (- \frac{36}{5} + \frac{y^{2}}{5}) + \frac{y^{2}}{5} \cdot (- \frac{36}{5} + \frac{y^{2}}{5}) + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Apply the distributive property.

$- \frac{36}{5} \cdot (- \frac{36}{5}) - \frac{36}{5} \cdot \frac{y^{2}}{5} + \frac{y^{2}}{5} \cdot (- \frac{36}{5} + \frac{y^{2}}{5}) + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Apply the distributive property.

$- \frac{36}{5} \cdot (- \frac{36}{5}) - \frac{36}{5} \cdot \frac{y^{2}}{5} + \frac{y^{2}}{5} \cdot (- \frac{36}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$- \frac{36}{5} \cdot (- \frac{36}{5}) - \frac{36}{5} \cdot \frac{y^{2}}{5} + \frac{y^{2}}{5} \cdot (- \frac{36}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Simplify and combine like terms.

Simplify each term.

Multiply $- \frac{36}{5} (- \frac{36}{5})$ .

Multiply $- 1$ by $- 1$ .

$1 (\frac{36}{5}) \cdot \frac{36}{5} - \frac{36}{5} \cdot \frac{y^{2}}{5} + \frac{y^{2}}{5} \cdot (- \frac{36}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Multiply $\frac{36}{5}$ by $1$ .

$\frac{36}{5} \cdot \frac{36}{5} - \frac{36}{5} \cdot \frac{y^{2}}{5} + \frac{y^{2}}{5} \cdot (- \frac{36}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Multiply $\frac{36}{5}$ by $\frac{36}{5}$ .

$\frac{36 \cdot 36}{5 \cdot 5} - \frac{36}{5} \cdot \frac{y^{2}}{5} + \frac{y^{2}}{5} \cdot (- \frac{36}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Multiply $36$ by $36$ .

$\frac{1296}{5 \cdot 5} - \frac{36}{5} \cdot \frac{y^{2}}{5} + \frac{y^{2}}{5} \cdot (- \frac{36}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Multiply $5$ by $5$ .

$\frac{1296}{25} - \frac{36}{5} \cdot \frac{y^{2}}{5} + \frac{y^{2}}{5} \cdot (- \frac{36}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$\frac{1296}{25} - \frac{36}{5} \cdot \frac{y^{2}}{5} + \frac{y^{2}}{5} \cdot (- \frac{36}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Multiply $- \frac{36}{5} \cdot \frac{y^{2}}{5}$ .

Multiply $\frac{y^{2}}{5}$ by $\frac{36}{5}$ .

$\frac{1296}{25} - \frac{y^{2} \cdot 36}{5 \cdot 5} + \frac{y^{2}}{5} \cdot (- \frac{36}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Multiply $5$ by $5$ .

$\frac{1296}{25} - \frac{y^{2} \cdot 36}{25} + \frac{y^{2}}{5} \cdot (- \frac{36}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$\frac{1296}{25} - \frac{y^{2} \cdot 36}{25} + \frac{y^{2}}{5} \cdot (- \frac{36}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Move $36$ to the left of $y^{2}$ .

$\frac{1296}{25} - \frac{36 \cdot y^{2}}{25} + \frac{y^{2}}{5} \cdot (- \frac{36}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Multiply $\frac{y^{2}}{5} (- \frac{36}{5})$ .

Multiply $\frac{y^{2}}{5}$ by $\frac{36}{5}$ .

$\frac{1296}{25} - \frac{36 y^{2}}{25} - \frac{y^{2} \cdot 36}{5 \cdot 5} + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Multiply $5$ by $5$ .

$\frac{1296}{25} - \frac{36 y^{2}}{25} - \frac{y^{2} \cdot 36}{25} + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$\frac{1296}{25} - \frac{36 y^{2}}{25} - \frac{y^{2} \cdot 36}{25} + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Move $36$ to the left of $y^{2}$ .

$\frac{1296}{25} - \frac{36 y^{2}}{25} - \frac{36 \cdot y^{2}}{25} + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Combine.

$\frac{1296}{25} - \frac{36 y^{2}}{25} - \frac{36 y^{2}}{25} + \frac{y^{2} y^{2}}{5 \cdot 5} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

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

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

$\frac{1296}{25} - \frac{36 y^{2}}{25} - \frac{36 y^{2}}{25} + \frac{y^{2 + 2}}{5 \cdot 5} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Add $2$ and $2$ .

$\frac{1296}{25} - \frac{36 y^{2}}{25} - \frac{36 y^{2}}{25} + \frac{y^{4}}{5 \cdot 5} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$\frac{1296}{25} - \frac{36 y^{2}}{25} - \frac{36 y^{2}}{25} + \frac{y^{4}}{5 \cdot 5} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Multiply $5$ by $5$ .

$\frac{1296}{25} - \frac{36 y^{2}}{25} - \frac{36 y^{2}}{25} + \frac{y^{4}}{25} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$\frac{1296}{25} - \frac{36 y^{2}}{25} - \frac{36 y^{2}}{25} + \frac{y^{4}}{25} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Subtract $\frac{36 y^{2}}{25}$ from $- \frac{36 y^{2}}{25}$ .

$\frac{1296}{25} - 2 \frac{36 y^{2}}{25} + \frac{y^{4}}{25} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$\frac{1296}{25} - 2 \frac{36 y^{2}}{25} + \frac{y^{4}}{25} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Simplify each term.

Multiply $- 2 \frac{36 y^{2}}{25}$ .

Combine $- 2$ and $\frac{36 y^{2}}{25}$ .

$\frac{1296}{25} + \frac{- 2 (36 y^{2})}{25} + \frac{y^{4}}{25} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Multiply $36$ by $- 2$ .

$\frac{1296}{25} + \frac{- 72 y^{2}}{25} + \frac{y^{4}}{25} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$\frac{1296}{25} + \frac{- 72 y^{2}}{25} + \frac{y^{4}}{25} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Move the negative in front of the fraction.

$\frac{1296}{25} - \frac{72 y^{2}}{25} + \frac{y^{4}}{25} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$\frac{1296}{25} - \frac{72 y^{2}}{25} + \frac{y^{4}}{25} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$\frac{1296}{25} - \frac{72 y^{2}}{25} + \frac{y^{4}}{25} + y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

To write $y^{2}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$\frac{1296}{25} + \frac{y^{4}}{25} - \frac{72 y^{2}}{25} + y^{2} \cdot \frac{25}{25} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Simplify terms.

Combine $y^{2}$ and $\frac{25}{25}$ .

$\frac{1296}{25} + \frac{y^{4}}{25} - \frac{72 y^{2}}{25} + \frac{y^{2} \cdot 25}{25} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Combine the numerators over the common denominator.

$\frac{1296}{25} + \frac{y^{4}}{25} + \frac{- 72 y^{2} + y^{2} \cdot 25}{25} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Combine the numerators over the common denominator.

$\frac{1296 + y^{4} - 72 y^{2} + y^{2} \cdot 25}{25} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$\frac{1296 + y^{4} - 72 y^{2} + y^{2} \cdot 25}{25} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Move $25$ to the left of $y^{2}$ .

$\frac{1296 + y^{4} - 72 y^{2} + 25 y^{2}}{25} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Simplify by adding terms.

Add $- 72 y^{2}$ and $25 y^{2}$ .

$\frac{1296 + y^{4} - 47 y^{2}}{25} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Reorder terms.

$\frac{y^{4} - 47 y^{2} + 1296}{25} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$\frac{y^{4} - 47 y^{2} + 1296}{25} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$\frac{y^{4} - 47 y^{2} + 1296}{25} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$\frac{y^{4} - 47 y^{2} + 1296}{25} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$\frac{y^{4} - 47 y^{2} + 1296}{25} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Step 3

Solve for $y$ in $\frac{y^{4} - 47 y^{2} + 1296}{25} = 36$ .

Multiply both sides by $25$ .

$\frac{y^{4} - 47 y^{2} + 1296}{25} \cdot 25 = 36 \cdot 25$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Simplify.

Simplify the left side.

Cancel the common factor of $25$ .

Cancel the common factor.

$\frac{y^{4} - 47 y^{2} + 1296}{25} \cdot 25 = 36 \cdot 25$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Rewrite the expression.

$y^{4} - 47 y^{2} + 1296 = 36 \cdot 25$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$y^{4} - 47 y^{2} + 1296 = 36 \cdot 25$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$y^{4} - 47 y^{2} + 1296 = 36 \cdot 25$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Simplify the right side.

Multiply $36$ by $25$ .

$y^{4} - 47 y^{2} + 1296 = 900$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$y^{4} - 47 y^{2} + 1296 = 900$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$y^{4} - 47 y^{2} + 1296 = 900$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Solve for $y$ .

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

$u^{2} - 47 u + 1296 = 900$

$u = y^{2}$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Subtract $900$ from both sides of the equation.

$u^{2} - 47 u + 1296 - 900 = 0$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Subtract $900$ from $1296$ .

$u^{2} - 47 u + 396 = 0$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Factor $u^{2} - 47 u + 396$ 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 $396$ and whose sum is $- 47$ .

$- 36, - 11$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Write the factored form using these integers.

$(u - 36) (u - 11) = 0$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$(u - 36) (u - 11) = 0$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

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

$u - 36 = 0$

$u - 11 = 0$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Set $u - 36$ equal to $0$ and solve for $u$ .

Set $u - 36$ equal to $0$ .

$u - 36 = 0$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Add $36$ to both sides of the equation.

$u = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$u = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Set $u - 11$ equal to $0$ and solve for $u$ .

Set $u - 11$ equal to $0$ .

$u - 11 = 0$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Add $11$ to both sides of the equation.

$u = 11$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$u = 11$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

The final solution is all the values that make $(u - 36) (u - 11) = 0$ true.

$u = 36, 11$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

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

$y^{2} = 36$

$y^{2} = 11$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Solve the first equation for $y$ .

$y^{2} = 36$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Solve the equation for $y$ .

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

$y = \pm \sqrt{36}$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Simplify $\pm \sqrt{36}$ .

Rewrite $36$ as $6^{2}$ .

$y = \pm \sqrt{6^{2}}$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

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

$y = \pm 6$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$y = \pm 6$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

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.

$y = 6$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

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

$y = - 6$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

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

$y = 6, - 6$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$y = 6, - 6$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$y = 6, - 6$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Solve the second equation for $y$ .

$y^{2} = 11$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Solve the equation for $y$ .

Remove parentheses.

$y^{2} = 11$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

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

$y = \pm \sqrt{11}$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

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.

$y = \sqrt{11}$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

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

$y = - \sqrt{11}$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

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

$y = \sqrt{11}, - \sqrt{11}$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$y = \sqrt{11}, - \sqrt{11}$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$y = \sqrt{11}, - \sqrt{11}$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

The solution to $y^{4} - 47 y^{2} + 1296 = 900$ is $y = 6, - 6, \sqrt{11}, - \sqrt{11}$ .

$y = 6, - 6, \sqrt{11}, - \sqrt{11}$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$y = 6, - 6, \sqrt{11}, - \sqrt{11}$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

$y = 6, - 6, \sqrt{11}, - \sqrt{11}$

$x = - \frac{36}{5} + \frac{y^{2}}{5}$

Step 4

Replace all occurrences of $y$ with $6$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{36}{5} + \frac{y^{2}}{5}$ with $6$ .

$x = - \frac{36}{5} + \frac{{(6)}^{2}}{5}$

$y = 6$

Simplify the right side.

Simplify $- \frac{36}{5} + \frac{{(6)}^{2}}{5}$ .

Combine the numerators over the common denominator.

$x = \frac{- 36 + {(6)}^{2}}{5}$

$y = 6$

Simplify the expression.

Raise $6$ to the power of $2$ .

$x = \frac{- 36 + 36}{5}$

$y = 6$

Add $- 36$ and $36$ .

$x = \frac{0}{5}$

$y = 6$

Divide $0$ by $5$ .

$x = 0$

$y = 6$

$x = 0$

$y = 6$

$x = 0$

$y = 6$

$x = 0$

$y = 6$

$x = 0$

$y = 6$

Step 5

Replace all occurrences of $y$ with $- 6$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{36}{5} + \frac{y^{2}}{5}$ with $- 6$ .

$x = - \frac{36}{5} + \frac{{(- 6)}^{2}}{5}$

$y = - 6$

Simplify the right side.

Simplify $- \frac{36}{5} + \frac{{(- 6)}^{2}}{5}$ .

Combine the numerators over the common denominator.

$x = \frac{- 36 + {(- 6)}^{2}}{5}$

$y = - 6$

Simplify the expression.

Raise $- 6$ to the power of $2$ .

$x = \frac{- 36 + 36}{5}$

$y = - 6$

Add $- 36$ and $36$ .

$x = \frac{0}{5}$

$y = - 6$

Divide $0$ by $5$ .

$x = 0$

$y = - 6$

$x = 0$

$y = - 6$

$x = 0$

$y = - 6$

$x = 0$

$y = - 6$

$x = 0$

$y = - 6$

Step 6

Replace all occurrences of $y$ with $6$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{36}{5} + \frac{y^{2}}{5}$ with $6$ .

$x = - \frac{36}{5} + \frac{{(6)}^{2}}{5}$

$y = 6$

Simplify the right side.

Simplify $- \frac{36}{5} + \frac{{(6)}^{2}}{5}$ .

Combine the numerators over the common denominator.

$x = \frac{- 36 + {(6)}^{2}}{5}$

$y = 6$

Simplify the expression.

Raise $6$ to the power of $2$ .

$x = \frac{- 36 + 36}{5}$

$y = 6$

Add $- 36$ and $36$ .

$x = \frac{0}{5}$

$y = 6$

Divide $0$ by $5$ .

$x = 0$

$y = 6$

$x = 0$

$y = 6$

$x = 0$

$y = 6$

$x = 0$

$y = 6$

$x = 0$

$y = 6$

Step 7

Replace all occurrences of $y$ with $- 6$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{36}{5} + \frac{y^{2}}{5}$ with $- 6$ .

$x = - \frac{36}{5} + \frac{{(- 6)}^{2}}{5}$

$y = - 6$

Simplify the right side.

Simplify $- \frac{36}{5} + \frac{{(- 6)}^{2}}{5}$ .

Combine the numerators over the common denominator.

$x = \frac{- 36 + {(- 6)}^{2}}{5}$

$y = - 6$

Simplify the expression.

Raise $- 6$ to the power of $2$ .

$x = \frac{- 36 + 36}{5}$

$y = - 6$

Add $- 36$ and $36$ .

$x = \frac{0}{5}$

$y = - 6$

Divide $0$ by $5$ .

$x = 0$

$y = - 6$

$x = 0$

$y = - 6$

$x = 0$

$y = - 6$

$x = 0$

$y = - 6$

$x = 0$

$y = - 6$

Step 8

Replace all occurrences of $y$ with $\sqrt{11}$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{36}{5} + \frac{y^{2}}{5}$ with $\sqrt{11}$ .

$x = - \frac{36}{5} + \frac{{(\sqrt{11})}^{2}}{5}$

$y = \sqrt{11}$

Simplify the right side.

Simplify $- \frac{36}{5} + \frac{{(\sqrt{11})}^{2}}{5}$ .

Combine the numerators over the common denominator.

$x = \frac{- 36 + {(\sqrt{11})}^{2}}{5}$

$y = \sqrt{11}$

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$x = \frac{- 36 + {(11^{\frac{1}{2}})}^{2}}{5}$

$y = \sqrt{11}$

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

$x = \frac{- 36 + 11^{\frac{1}{2} \cdot 2}}{5}$

$y = \sqrt{11}$

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

$x = \frac{- 36 + 11^{\frac{2}{2}}}{5}$

$y = \sqrt{11}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \frac{- 36 + 11^{\frac{2}{2}}}{5}$

$y = \sqrt{11}$

Rewrite the expression.

$x = \frac{- 36 + 11}{5}$

$y = \sqrt{11}$

$x = \frac{- 36 + 11}{5}$

$y = \sqrt{11}$

Evaluate the exponent.

$x = \frac{- 36 + 11}{5}$

$y = \sqrt{11}$

$x = \frac{- 36 + 11}{5}$

$y = \sqrt{11}$

Simplify the expression.

Add $- 36$ and $11$ .

$x = \frac{- 25}{5}$

$y = \sqrt{11}$

Divide $- 25$ by $5$ .

$x = - 5$

$y = \sqrt{11}$

$x = - 5$

$y = \sqrt{11}$

$x = - 5$

$y = \sqrt{11}$

$x = - 5$

$y = \sqrt{11}$

$x = - 5$

$y = \sqrt{11}$

Step 9

Replace all occurrences of $y$ with $6$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{36}{5} + \frac{y^{2}}{5}$ with $6$ .

$x = - \frac{36}{5} + \frac{{(6)}^{2}}{5}$

$y = 6$

Simplify the right side.

Simplify $- \frac{36}{5} + \frac{{(6)}^{2}}{5}$ .

Combine the numerators over the common denominator.

$x = \frac{- 36 + {(6)}^{2}}{5}$

$y = 6$

Simplify the expression.

Raise $6$ to the power of $2$ .

$x = \frac{- 36 + 36}{5}$

$y = 6$

Add $- 36$ and $36$ .

$x = \frac{0}{5}$

$y = 6$

Divide $0$ by $5$ .

$x = 0$

$y = 6$

$x = 0$

$y = 6$

$x = 0$

$y = 6$

$x = 0$

$y = 6$

$x = 0$

$y = 6$

Step 10

Replace all occurrences of $y$ with $- 6$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{36}{5} + \frac{y^{2}}{5}$ with $- 6$ .

$x = - \frac{36}{5} + \frac{{(- 6)}^{2}}{5}$

$y = - 6$

Simplify the right side.

Simplify $- \frac{36}{5} + \frac{{(- 6)}^{2}}{5}$ .

Combine the numerators over the common denominator.

$x = \frac{- 36 + {(- 6)}^{2}}{5}$

$y = - 6$

Simplify the expression.

Raise $- 6$ to the power of $2$ .

$x = \frac{- 36 + 36}{5}$

$y = - 6$

Add $- 36$ and $36$ .

$x = \frac{0}{5}$

$y = - 6$

Divide $0$ by $5$ .

$x = 0$

$y = - 6$

$x = 0$

$y = - 6$

$x = 0$

$y = - 6$

$x = 0$

$y = - 6$

$x = 0$

$y = - 6$

Step 11

Replace all occurrences of $y$ with $\sqrt{11}$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{36}{5} + \frac{y^{2}}{5}$ with $\sqrt{11}$ .

$x = - \frac{36}{5} + \frac{{(\sqrt{11})}^{2}}{5}$

$y = \sqrt{11}$

Simplify the right side.

Simplify $- \frac{36}{5} + \frac{{(\sqrt{11})}^{2}}{5}$ .

Combine the numerators over the common denominator.

$x = \frac{- 36 + {(\sqrt{11})}^{2}}{5}$

$y = \sqrt{11}$

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$x = \frac{- 36 + {(11^{\frac{1}{2}})}^{2}}{5}$

$y = \sqrt{11}$

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

$x = \frac{- 36 + 11^{\frac{1}{2} \cdot 2}}{5}$

$y = \sqrt{11}$

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

$x = \frac{- 36 + 11^{\frac{2}{2}}}{5}$

$y = \sqrt{11}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \frac{- 36 + 11^{\frac{2}{2}}}{5}$

$y = \sqrt{11}$

Rewrite the expression.

$x = \frac{- 36 + 11}{5}$

$y = \sqrt{11}$

$x = \frac{- 36 + 11}{5}$

$y = \sqrt{11}$

Evaluate the exponent.

$x = \frac{- 36 + 11}{5}$

$y = \sqrt{11}$

$x = \frac{- 36 + 11}{5}$

$y = \sqrt{11}$

Simplify the expression.

Add $- 36$ and $11$ .

$x = \frac{- 25}{5}$

$y = \sqrt{11}$

Divide $- 25$ by $5$ .

$x = - 5$

$y = \sqrt{11}$

$x = - 5$

$y = \sqrt{11}$

$x = - 5$

$y = \sqrt{11}$

$x = - 5$

$y = \sqrt{11}$

$x = - 5$

$y = \sqrt{11}$

Step 12

Replace all occurrences of $y$ with $- \sqrt{11}$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{36}{5} + \frac{y^{2}}{5}$ with $- \sqrt{11}$ .

$x = - \frac{36}{5} + \frac{{(- \sqrt{11})}^{2}}{5}$

$y = - \sqrt{11}$

Simplify the right side.

Simplify $- \frac{36}{5} + \frac{{(- \sqrt{11})}^{2}}{5}$ .

Combine the numerators over the common denominator.

$x = \frac{- 36 + {(- \sqrt{11})}^{2}}{5}$

$y = - \sqrt{11}$

Simplify each term.

Apply the product rule to $- \sqrt{11}$ .

$x = \frac{- 36 + {(- 1)}^{2} {\sqrt{11}}^{2}}{5}$

$y = - \sqrt{11}$

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

$x = \frac{- 36 + 1 {\sqrt{11}}^{2}}{5}$

$y = - \sqrt{11}$

Multiply ${\sqrt{11}}^{2}$ by $1$ .

$x = \frac{- 36 + {\sqrt{11}}^{2}}{5}$

$y = - \sqrt{11}$

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$x = \frac{- 36 + {(11^{\frac{1}{2}})}^{2}}{5}$

$y = - \sqrt{11}$

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

$x = \frac{- 36 + 11^{\frac{1}{2} \cdot 2}}{5}$

$y = - \sqrt{11}$

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

$x = \frac{- 36 + 11^{\frac{2}{2}}}{5}$

$y = - \sqrt{11}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \frac{- 36 + 11^{\frac{2}{2}}}{5}$

$y = - \sqrt{11}$

Rewrite the expression.

$x = \frac{- 36 + 11}{5}$

$y = - \sqrt{11}$

$x = \frac{- 36 + 11}{5}$

$y = - \sqrt{11}$

Evaluate the exponent.

$x = \frac{- 36 + 11}{5}$

$y = - \sqrt{11}$

$x = \frac{- 36 + 11}{5}$

$y = - \sqrt{11}$

$x = \frac{- 36 + 11}{5}$

$y = - \sqrt{11}$

Simplify the expression.

Add $- 36$ and $11$ .

$x = \frac{- 25}{5}$

$y = - \sqrt{11}$

Divide $- 25$ by $5$ .

$x = - 5$

$y = - \sqrt{11}$

$x = - 5$

$y = - \sqrt{11}$

$x = - 5$

$y = - \sqrt{11}$

$x = - 5$

$y = - \sqrt{11}$

$x = - 5$

$y = - \sqrt{11}$

Step 13

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

$(0, 6)$

$(0, - 6)$

$(- 5, \sqrt{11})$

$(- 5, - \sqrt{11})$

Step 14

The result can be shown in multiple forms.

Point Form:

$(0, 6), (0, - 6), (- 5, \sqrt{11}), (- 5, - \sqrt{11})$

Equation Form:

$x = 0, y = 6$

$x = 0, y = - 6$

$x = - 5, y = \sqrt{11}$

$x = - 5, y = - \sqrt{11}$

Step 15

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