Solve by Substitution x^2+y^2=49 , y^2-5x=49

$x^{2} + y^{2} = 49$ , $y^{2} - 5 x = 49$

Step 1

Solve for $x$ in $y^{2} - 5 x = 49$ .

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

$- 5 x = 49 - y^{2}$

$x^{2} + y^{2} = 49$

Divide each term in $- 5 x = 49 - y^{2}$ by $- 5$ and simplify.

Divide each term in $- 5 x = 49 - y^{2}$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{49}{- 5} + \frac{- y^{2}}{- 5}$

$x^{2} + y^{2} = 49$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{49}{- 5} + \frac{- y^{2}}{- 5}$

$x^{2} + y^{2} = 49$

Divide $x$ by $1$ .

$x = \frac{49}{- 5} + \frac{- y^{2}}{- 5}$

$x^{2} + y^{2} = 49$

$x = \frac{49}{- 5} + \frac{- y^{2}}{- 5}$

$x^{2} + y^{2} = 49$

$x = \frac{49}{- 5} + \frac{- y^{2}}{- 5}$

$x^{2} + y^{2} = 49$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{49}{5} + \frac{- y^{2}}{- 5}$

$x^{2} + y^{2} = 49$

Dividing two negative values results in a positive value.

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$x^{2} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$x^{2} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$x^{2} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$x^{2} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$x^{2} + y^{2} = 49$

Step 2

Replace all occurrences of $x$ with $- \frac{49}{5} + \frac{y^{2}}{5}$ in each equation.

Replace all occurrences of $x$ in $x^{2} + y^{2} = 49$ with $- \frac{49}{5} + \frac{y^{2}}{5}$ .

${(- \frac{49}{5} + \frac{y^{2}}{5})}^{2} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Simplify the left side.

Simplify ${(- \frac{49}{5} + \frac{y^{2}}{5})}^{2} + y^{2}$ .

Simplify each term.

Rewrite ${(- \frac{49}{5} + \frac{y^{2}}{5})}^{2}$ as $(- \frac{49}{5} + \frac{y^{2}}{5}) (- \frac{49}{5} + \frac{y^{2}}{5})$ .

$(- \frac{49}{5} + \frac{y^{2}}{5}) (- \frac{49}{5} + \frac{y^{2}}{5}) + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Expand $(- \frac{49}{5} + \frac{y^{2}}{5}) (- \frac{49}{5} + \frac{y^{2}}{5})$ using the FOIL Method.

Apply the distributive property.

$- \frac{49}{5} \cdot (- \frac{49}{5} + \frac{y^{2}}{5}) + \frac{y^{2}}{5} \cdot (- \frac{49}{5} + \frac{y^{2}}{5}) + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Apply the distributive property.

$- \frac{49}{5} \cdot (- \frac{49}{5}) - \frac{49}{5} \cdot \frac{y^{2}}{5} + \frac{y^{2}}{5} \cdot (- \frac{49}{5} + \frac{y^{2}}{5}) + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Apply the distributive property.

$- \frac{49}{5} \cdot (- \frac{49}{5}) - \frac{49}{5} \cdot \frac{y^{2}}{5} + \frac{y^{2}}{5} \cdot (- \frac{49}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$- \frac{49}{5} \cdot (- \frac{49}{5}) - \frac{49}{5} \cdot \frac{y^{2}}{5} + \frac{y^{2}}{5} \cdot (- \frac{49}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Simplify and combine like terms.

Simplify each term.

Multiply $- \frac{49}{5} (- \frac{49}{5})$ .

Multiply $- 1$ by $- 1$ .

$1 (\frac{49}{5}) \cdot \frac{49}{5} - \frac{49}{5} \cdot \frac{y^{2}}{5} + \frac{y^{2}}{5} \cdot (- \frac{49}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Multiply $\frac{49}{5}$ by $1$ .

$\frac{49}{5} \cdot \frac{49}{5} - \frac{49}{5} \cdot \frac{y^{2}}{5} + \frac{y^{2}}{5} \cdot (- \frac{49}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Multiply $\frac{49}{5}$ by $\frac{49}{5}$ .

$\frac{49 \cdot 49}{5 \cdot 5} - \frac{49}{5} \cdot \frac{y^{2}}{5} + \frac{y^{2}}{5} \cdot (- \frac{49}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Multiply $49$ by $49$ .

$\frac{2401}{5 \cdot 5} - \frac{49}{5} \cdot \frac{y^{2}}{5} + \frac{y^{2}}{5} \cdot (- \frac{49}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Multiply $5$ by $5$ .

$\frac{2401}{25} - \frac{49}{5} \cdot \frac{y^{2}}{5} + \frac{y^{2}}{5} \cdot (- \frac{49}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$\frac{2401}{25} - \frac{49}{5} \cdot \frac{y^{2}}{5} + \frac{y^{2}}{5} \cdot (- \frac{49}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Multiply $- \frac{49}{5} \cdot \frac{y^{2}}{5}$ .

Multiply $\frac{y^{2}}{5}$ by $\frac{49}{5}$ .

$\frac{2401}{25} - \frac{y^{2} \cdot 49}{5 \cdot 5} + \frac{y^{2}}{5} \cdot (- \frac{49}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Multiply $5$ by $5$ .

$\frac{2401}{25} - \frac{y^{2} \cdot 49}{25} + \frac{y^{2}}{5} \cdot (- \frac{49}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$\frac{2401}{25} - \frac{y^{2} \cdot 49}{25} + \frac{y^{2}}{5} \cdot (- \frac{49}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Move $49$ to the left of $y^{2}$ .

$\frac{2401}{25} - \frac{49 \cdot y^{2}}{25} + \frac{y^{2}}{5} \cdot (- \frac{49}{5}) + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Multiply $\frac{y^{2}}{5} (- \frac{49}{5})$ .

Multiply $\frac{y^{2}}{5}$ by $\frac{49}{5}$ .

$\frac{2401}{25} - \frac{49 y^{2}}{25} - \frac{y^{2} \cdot 49}{5 \cdot 5} + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Multiply $5$ by $5$ .

$\frac{2401}{25} - \frac{49 y^{2}}{25} - \frac{y^{2} \cdot 49}{25} + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$\frac{2401}{25} - \frac{49 y^{2}}{25} - \frac{y^{2} \cdot 49}{25} + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Move $49$ to the left of $y^{2}$ .

$\frac{2401}{25} - \frac{49 y^{2}}{25} - \frac{49 \cdot y^{2}}{25} + \frac{y^{2}}{5} \cdot \frac{y^{2}}{5} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Combine.

$\frac{2401}{25} - \frac{49 y^{2}}{25} - \frac{49 y^{2}}{25} + \frac{y^{2} y^{2}}{5 \cdot 5} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

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

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

$\frac{2401}{25} - \frac{49 y^{2}}{25} - \frac{49 y^{2}}{25} + \frac{y^{2 + 2}}{5 \cdot 5} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Add $2$ and $2$ .

$\frac{2401}{25} - \frac{49 y^{2}}{25} - \frac{49 y^{2}}{25} + \frac{y^{4}}{5 \cdot 5} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$\frac{2401}{25} - \frac{49 y^{2}}{25} - \frac{49 y^{2}}{25} + \frac{y^{4}}{5 \cdot 5} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Multiply $5$ by $5$ .

$\frac{2401}{25} - \frac{49 y^{2}}{25} - \frac{49 y^{2}}{25} + \frac{y^{4}}{25} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$\frac{2401}{25} - \frac{49 y^{2}}{25} - \frac{49 y^{2}}{25} + \frac{y^{4}}{25} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Subtract $\frac{49 y^{2}}{25}$ from $- \frac{49 y^{2}}{25}$ .

$\frac{2401}{25} - 2 \frac{49 y^{2}}{25} + \frac{y^{4}}{25} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$\frac{2401}{25} - 2 \frac{49 y^{2}}{25} + \frac{y^{4}}{25} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Simplify each term.

Multiply $- 2 \frac{49 y^{2}}{25}$ .

Combine $- 2$ and $\frac{49 y^{2}}{25}$ .

$\frac{2401}{25} + \frac{- 2 (49 y^{2})}{25} + \frac{y^{4}}{25} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Multiply $49$ by $- 2$ .

$\frac{2401}{25} + \frac{- 98 y^{2}}{25} + \frac{y^{4}}{25} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$\frac{2401}{25} + \frac{- 98 y^{2}}{25} + \frac{y^{4}}{25} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Move the negative in front of the fraction.

$\frac{2401}{25} - \frac{98 y^{2}}{25} + \frac{y^{4}}{25} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$\frac{2401}{25} - \frac{98 y^{2}}{25} + \frac{y^{4}}{25} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$\frac{2401}{25} - \frac{98 y^{2}}{25} + \frac{y^{4}}{25} + y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

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

$\frac{2401}{25} + \frac{y^{4}}{25} - \frac{98 y^{2}}{25} + y^{2} \cdot \frac{25}{25} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Simplify terms.

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

$\frac{2401}{25} + \frac{y^{4}}{25} - \frac{98 y^{2}}{25} + \frac{y^{2} \cdot 25}{25} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Combine the numerators over the common denominator.

$\frac{2401}{25} + \frac{y^{4}}{25} + \frac{- 98 y^{2} + y^{2} \cdot 25}{25} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Combine the numerators over the common denominator.

$\frac{2401 + y^{4} - 98 y^{2} + y^{2} \cdot 25}{25} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$\frac{2401 + y^{4} - 98 y^{2} + y^{2} \cdot 25}{25} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

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

$\frac{2401 + y^{4} - 98 y^{2} + 25 y^{2}}{25} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Simplify by adding terms.

Add $- 98 y^{2}$ and $25 y^{2}$ .

$\frac{2401 + y^{4} - 73 y^{2}}{25} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Reorder terms.

$\frac{y^{4} - 73 y^{2} + 2401}{25} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$\frac{y^{4} - 73 y^{2} + 2401}{25} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$\frac{y^{4} - 73 y^{2} + 2401}{25} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$\frac{y^{4} - 73 y^{2} + 2401}{25} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$\frac{y^{4} - 73 y^{2} + 2401}{25} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Step 3

Solve for $y$ in $\frac{y^{4} - 73 y^{2} + 2401}{25} = 49$ .

Multiply both sides by $25$ .

$\frac{y^{4} - 73 y^{2} + 2401}{25} \cdot 25 = 49 \cdot 25$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Simplify.

Simplify the left side.

Cancel the common factor of $25$ .

Cancel the common factor.

$\frac{y^{4} - 73 y^{2} + 2401}{25} \cdot 25 = 49 \cdot 25$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Rewrite the expression.

$y^{4} - 73 y^{2} + 2401 = 49 \cdot 25$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$y^{4} - 73 y^{2} + 2401 = 49 \cdot 25$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$y^{4} - 73 y^{2} + 2401 = 49 \cdot 25$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Simplify the right side.

Multiply $49$ by $25$ .

$y^{4} - 73 y^{2} + 2401 = 1225$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$y^{4} - 73 y^{2} + 2401 = 1225$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$y^{4} - 73 y^{2} + 2401 = 1225$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Solve for $y$ .

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

$u^{2} - 73 u + 2401 = 1225$

$u = y^{2}$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Subtract $1225$ from both sides of the equation.

$u^{2} - 73 u + 2401 - 1225 = 0$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Subtract $1225$ from $2401$ .

$u^{2} - 73 u + 1176 = 0$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Factor $u^{2} - 73 u + 1176$ 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 $1176$ and whose sum is $- 73$ .

$- 49, - 24$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Write the factored form using these integers.

$(u - 49) (u - 24) = 0$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$(u - 49) (u - 24) = 0$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

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

$u - 49 = 0$

$u - 24 = 0$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Set $u - 49$ equal to $0$ and solve for $u$ .

Set $u - 49$ equal to $0$ .

$u - 49 = 0$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Add $49$ to both sides of the equation.

$u = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$u = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Set $u - 24$ equal to $0$ and solve for $u$ .

Set $u - 24$ equal to $0$ .

$u - 24 = 0$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Add $24$ to both sides of the equation.

$u = 24$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$u = 24$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

The final solution is all the values that make $(u - 49) (u - 24) = 0$ true.

$u = 49, 24$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

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

$y^{2} = 49$

$y^{2} = 24$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Solve the first equation for $y$ .

$y^{2} = 49$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Solve the equation for $y$ .

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

$y = \pm \sqrt{49}$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Simplify $\pm \sqrt{49}$ .

Rewrite $49$ as $7^{2}$ .

$y = \pm \sqrt{7^{2}}$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

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

$y = \pm 7$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$y = \pm 7$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

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.

$y = 7$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

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

$y = - 7$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

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

$y = 7, - 7$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$y = 7, - 7$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$y = 7, - 7$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Solve the second equation for $y$ .

$y^{2} = 24$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Solve the equation for $y$ .

Remove parentheses.

$y^{2} = 24$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

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

$y = \pm \sqrt{24}$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Simplify $\pm \sqrt{24}$ .

Rewrite $24$ as $2^{2} \cdot 6$ .

Factor $4$ out of $24$ .

$y = \pm \sqrt{4 (6)}$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

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

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

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

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

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Pull terms out from under the radical.

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

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

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

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

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.

$y = 2 \sqrt{6}$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

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

$y = - 2 \sqrt{6}$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

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

$y = 2 \sqrt{6}, - 2 \sqrt{6}$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$y = 2 \sqrt{6}, - 2 \sqrt{6}$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$y = 2 \sqrt{6}, - 2 \sqrt{6}$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

The solution to $y^{4} - 73 y^{2} + 2401 = 1225$ is $y = 7, - 7, 2 \sqrt{6}, - 2 \sqrt{6}$ .

$y = 7, - 7, 2 \sqrt{6}, - 2 \sqrt{6}$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$y = 7, - 7, 2 \sqrt{6}, - 2 \sqrt{6}$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

$y = 7, - 7, 2 \sqrt{6}, - 2 \sqrt{6}$

$x = - \frac{49}{5} + \frac{y^{2}}{5}$

Step 4

Replace all occurrences of $y$ with $7$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{49}{5} + \frac{y^{2}}{5}$ with $7$ .

$x = - \frac{49}{5} + \frac{{(7)}^{2}}{5}$

$y = 7$

Simplify the right side.

Simplify $- \frac{49}{5} + \frac{{(7)}^{2}}{5}$ .

Combine the numerators over the common denominator.

$x = \frac{- 49 + {(7)}^{2}}{5}$

$y = 7$

Simplify the expression.

Raise $7$ to the power of $2$ .

$x = \frac{- 49 + 49}{5}$

$y = 7$

Add $- 49$ and $49$ .

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

$y = 7$

Divide $0$ by $5$ .

$x = 0$

$y = 7$

$x = 0$

$y = 7$

$x = 0$

$y = 7$

$x = 0$

$y = 7$

$x = 0$

$y = 7$

Step 5

Replace all occurrences of $y$ with $- 7$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{49}{5} + \frac{y^{2}}{5}$ with $- 7$ .

$x = - \frac{49}{5} + \frac{{(- 7)}^{2}}{5}$

$y = - 7$

Simplify the right side.

Simplify $- \frac{49}{5} + \frac{{(- 7)}^{2}}{5}$ .

Combine the numerators over the common denominator.

$x = \frac{- 49 + {(- 7)}^{2}}{5}$

$y = - 7$

Simplify the expression.

Raise $- 7$ to the power of $2$ .

$x = \frac{- 49 + 49}{5}$

$y = - 7$

Add $- 49$ and $49$ .

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

$y = - 7$

Divide $0$ by $5$ .

$x = 0$

$y = - 7$

$x = 0$

$y = - 7$

$x = 0$

$y = - 7$

$x = 0$

$y = - 7$

$x = 0$

$y = - 7$

Step 6

Replace all occurrences of $y$ with $7$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{49}{5} + \frac{y^{2}}{5}$ with $7$ .

$x = - \frac{49}{5} + \frac{{(7)}^{2}}{5}$

$y = 7$

Simplify the right side.

Simplify $- \frac{49}{5} + \frac{{(7)}^{2}}{5}$ .

Combine the numerators over the common denominator.

$x = \frac{- 49 + {(7)}^{2}}{5}$

$y = 7$

Simplify the expression.

Raise $7$ to the power of $2$ .

$x = \frac{- 49 + 49}{5}$

$y = 7$

Add $- 49$ and $49$ .

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

$y = 7$

Divide $0$ by $5$ .

$x = 0$

$y = 7$

$x = 0$

$y = 7$

$x = 0$

$y = 7$

$x = 0$

$y = 7$

$x = 0$

$y = 7$

Step 7

Replace all occurrences of $y$ with $- 7$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{49}{5} + \frac{y^{2}}{5}$ with $- 7$ .

$x = - \frac{49}{5} + \frac{{(- 7)}^{2}}{5}$

$y = - 7$

Simplify the right side.

Simplify $- \frac{49}{5} + \frac{{(- 7)}^{2}}{5}$ .

Combine the numerators over the common denominator.

$x = \frac{- 49 + {(- 7)}^{2}}{5}$

$y = - 7$

Simplify the expression.

Raise $- 7$ to the power of $2$ .

$x = \frac{- 49 + 49}{5}$

$y = - 7$

Add $- 49$ and $49$ .

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

$y = - 7$

Divide $0$ by $5$ .

$x = 0$

$y = - 7$

$x = 0$

$y = - 7$

$x = 0$

$y = - 7$

$x = 0$

$y = - 7$

$x = 0$

$y = - 7$

Step 8

Replace all occurrences of $y$ with $2 \sqrt{6}$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{49}{5} + \frac{y^{2}}{5}$ with $2 \sqrt{6}$ .

$x = - \frac{49}{5} + \frac{{(2 \sqrt{6})}^{2}}{5}$

$y = 2 \sqrt{6}$

Simplify the right side.

Simplify $- \frac{49}{5} + \frac{{(2 \sqrt{6})}^{2}}{5}$ .

Combine the numerators over the common denominator.

$x = \frac{- 49 + {(2 \sqrt{6})}^{2}}{5}$

$y = 2 \sqrt{6}$

Simplify each term.

Apply the product rule to $2 \sqrt{6}$ .

$x = \frac{- 49 + 2^{2} {\sqrt{6}}^{2}}{5}$

$y = 2 \sqrt{6}$

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

$x = \frac{- 49 + 4 {\sqrt{6}}^{2}}{5}$

$y = 2 \sqrt{6}$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$x = \frac{- 49 + 4 {(6^{\frac{1}{2}})}^{2}}{5}$

$y = 2 \sqrt{6}$

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

$x = \frac{- 49 + 4 \cdot 6^{\frac{1}{2} \cdot 2}}{5}$

$y = 2 \sqrt{6}$

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

$x = \frac{- 49 + 4 \cdot 6^{\frac{2}{2}}}{5}$

$y = 2 \sqrt{6}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \frac{- 49 + 4 \cdot 6^{\frac{2}{2}}}{5}$

$y = 2 \sqrt{6}$

Rewrite the expression.

$x = \frac{- 49 + 4 \cdot 6}{5}$

$y = 2 \sqrt{6}$

$x = \frac{- 49 + 4 \cdot 6}{5}$

$y = 2 \sqrt{6}$

Evaluate the exponent.

$x = \frac{- 49 + 4 \cdot 6}{5}$

$y = 2 \sqrt{6}$

$x = \frac{- 49 + 4 \cdot 6}{5}$

$y = 2 \sqrt{6}$

Multiply $4$ by $6$ .

$x = \frac{- 49 + 24}{5}$

$y = 2 \sqrt{6}$

$x = \frac{- 49 + 24}{5}$

$y = 2 \sqrt{6}$

Simplify the expression.

Add $- 49$ and $24$ .

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

$y = 2 \sqrt{6}$

Divide $- 25$ by $5$ .

$x = - 5$

$y = 2 \sqrt{6}$

$x = - 5$

$y = 2 \sqrt{6}$

$x = - 5$

$y = 2 \sqrt{6}$

$x = - 5$

$y = 2 \sqrt{6}$

$x = - 5$

$y = 2 \sqrt{6}$

Step 9

Replace all occurrences of $y$ with $7$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{49}{5} + \frac{y^{2}}{5}$ with $7$ .

$x = - \frac{49}{5} + \frac{{(7)}^{2}}{5}$

$y = 7$

Simplify the right side.

Simplify $- \frac{49}{5} + \frac{{(7)}^{2}}{5}$ .

Combine the numerators over the common denominator.

$x = \frac{- 49 + {(7)}^{2}}{5}$

$y = 7$

Simplify the expression.

Raise $7$ to the power of $2$ .

$x = \frac{- 49 + 49}{5}$

$y = 7$

Add $- 49$ and $49$ .

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

$y = 7$

Divide $0$ by $5$ .

$x = 0$

$y = 7$

$x = 0$

$y = 7$

$x = 0$

$y = 7$

$x = 0$

$y = 7$

$x = 0$

$y = 7$

Step 10

Replace all occurrences of $y$ with $- 7$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{49}{5} + \frac{y^{2}}{5}$ with $- 7$ .

$x = - \frac{49}{5} + \frac{{(- 7)}^{2}}{5}$

$y = - 7$

Simplify the right side.

Simplify $- \frac{49}{5} + \frac{{(- 7)}^{2}}{5}$ .

Combine the numerators over the common denominator.

$x = \frac{- 49 + {(- 7)}^{2}}{5}$

$y = - 7$

Simplify the expression.

Raise $- 7$ to the power of $2$ .

$x = \frac{- 49 + 49}{5}$

$y = - 7$

Add $- 49$ and $49$ .

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

$y = - 7$

Divide $0$ by $5$ .

$x = 0$

$y = - 7$

$x = 0$

$y = - 7$

$x = 0$

$y = - 7$

$x = 0$

$y = - 7$

$x = 0$

$y = - 7$

Step 11

Replace all occurrences of $y$ with $2 \sqrt{6}$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{49}{5} + \frac{y^{2}}{5}$ with $2 \sqrt{6}$ .

$x = - \frac{49}{5} + \frac{{(2 \sqrt{6})}^{2}}{5}$

$y = 2 \sqrt{6}$

Simplify the right side.

Simplify $- \frac{49}{5} + \frac{{(2 \sqrt{6})}^{2}}{5}$ .

Combine the numerators over the common denominator.

$x = \frac{- 49 + {(2 \sqrt{6})}^{2}}{5}$

$y = 2 \sqrt{6}$

Simplify each term.

Apply the product rule to $2 \sqrt{6}$ .

$x = \frac{- 49 + 2^{2} {\sqrt{6}}^{2}}{5}$

$y = 2 \sqrt{6}$

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

$x = \frac{- 49 + 4 {\sqrt{6}}^{2}}{5}$

$y = 2 \sqrt{6}$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$x = \frac{- 49 + 4 {(6^{\frac{1}{2}})}^{2}}{5}$

$y = 2 \sqrt{6}$

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

$x = \frac{- 49 + 4 \cdot 6^{\frac{1}{2} \cdot 2}}{5}$

$y = 2 \sqrt{6}$

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

$x = \frac{- 49 + 4 \cdot 6^{\frac{2}{2}}}{5}$

$y = 2 \sqrt{6}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \frac{- 49 + 4 \cdot 6^{\frac{2}{2}}}{5}$

$y = 2 \sqrt{6}$

Rewrite the expression.

$x = \frac{- 49 + 4 \cdot 6}{5}$

$y = 2 \sqrt{6}$

$x = \frac{- 49 + 4 \cdot 6}{5}$

$y = 2 \sqrt{6}$

Evaluate the exponent.

$x = \frac{- 49 + 4 \cdot 6}{5}$

$y = 2 \sqrt{6}$

$x = \frac{- 49 + 4 \cdot 6}{5}$

$y = 2 \sqrt{6}$

Multiply $4$ by $6$ .

$x = \frac{- 49 + 24}{5}$

$y = 2 \sqrt{6}$

$x = \frac{- 49 + 24}{5}$

$y = 2 \sqrt{6}$

Simplify the expression.

Add $- 49$ and $24$ .

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

$y = 2 \sqrt{6}$

Divide $- 25$ by $5$ .

$x = - 5$

$y = 2 \sqrt{6}$

$x = - 5$

$y = 2 \sqrt{6}$

$x = - 5$

$y = 2 \sqrt{6}$

$x = - 5$

$y = 2 \sqrt{6}$

$x = - 5$

$y = 2 \sqrt{6}$

Step 12

Replace all occurrences of $y$ with $- 2 \sqrt{6}$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{49}{5} + \frac{y^{2}}{5}$ with $- 2 \sqrt{6}$ .

$x = - \frac{49}{5} + \frac{{(- 2 \sqrt{6})}^{2}}{5}$

$y = - 2 \sqrt{6}$

Simplify the right side.

Simplify $- \frac{49}{5} + \frac{{(- 2 \sqrt{6})}^{2}}{5}$ .

Combine the numerators over the common denominator.

$x = \frac{- 49 + {(- 2 \sqrt{6})}^{2}}{5}$

$y = - 2 \sqrt{6}$

Simplify each term.

Apply the product rule to $- 2 \sqrt{6}$ .

$x = \frac{- 49 + {(- 2)}^{2} {\sqrt{6}}^{2}}{5}$

$y = - 2 \sqrt{6}$

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

$x = \frac{- 49 + 4 {\sqrt{6}}^{2}}{5}$

$y = - 2 \sqrt{6}$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$x = \frac{- 49 + 4 {(6^{\frac{1}{2}})}^{2}}{5}$

$y = - 2 \sqrt{6}$

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

$x = \frac{- 49 + 4 \cdot 6^{\frac{1}{2} \cdot 2}}{5}$

$y = - 2 \sqrt{6}$

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

$x = \frac{- 49 + 4 \cdot 6^{\frac{2}{2}}}{5}$

$y = - 2 \sqrt{6}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \frac{- 49 + 4 \cdot 6^{\frac{2}{2}}}{5}$

$y = - 2 \sqrt{6}$

Rewrite the expression.

$x = \frac{- 49 + 4 \cdot 6}{5}$

$y = - 2 \sqrt{6}$

$x = \frac{- 49 + 4 \cdot 6}{5}$

$y = - 2 \sqrt{6}$

Evaluate the exponent.

$x = \frac{- 49 + 4 \cdot 6}{5}$

$y = - 2 \sqrt{6}$

$x = \frac{- 49 + 4 \cdot 6}{5}$

$y = - 2 \sqrt{6}$

Multiply $4$ by $6$ .

$x = \frac{- 49 + 24}{5}$

$y = - 2 \sqrt{6}$

$x = \frac{- 49 + 24}{5}$

$y = - 2 \sqrt{6}$

Simplify the expression.

Add $- 49$ and $24$ .

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

$y = - 2 \sqrt{6}$

Divide $- 25$ by $5$ .

$x = - 5$

$y = - 2 \sqrt{6}$

$x = - 5$

$y = - 2 \sqrt{6}$

$x = - 5$

$y = - 2 \sqrt{6}$

$x = - 5$

$y = - 2 \sqrt{6}$

$x = - 5$

$y = - 2 \sqrt{6}$

Step 13

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

$(0, 7)$

$(0, - 7)$

$(- 5, 2 \sqrt{6})$

$(- 5, - 2 \sqrt{6})$

Step 14

The result can be shown in multiple forms.

Point Form:

$(0, 7), (0, - 7), (- 5, 2 \sqrt{6}), (- 5, - 2 \sqrt{6})$

Equation Form:

$x = 0, y = 7$

$x = 0, y = - 7$

$x = - 5, y = 2 \sqrt{6}$

$x = - 5, y = - 2 \sqrt{6}$

Step 15

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