Solve by Addition/Elimination x^2+y^2=125 , x^2-y^2=-75

$x^{2} + y^{2} = 125$ , $x^{2} - y^{2} = - 75$

Step 1

Simplify the left side.

Reorder $x^{2}$ and $y^{2}$ .

$y^{2} + x^{2} = 125$

$x^{2} - y^{2} = - 75$

$y^{2} + x^{2} = 125$

$x^{2} - y^{2} = - 75$

Step 2

Simplify the left side.

Reorder $x^{2}$ and $- y^{2}$ .

$- y^{2} + x^{2} = - 75$

$y^{2} + x^{2} = 125$

$- y^{2} + x^{2} = - 75$

$y^{2} + x^{2} = 125$

Step 3

Add the two equations together to eliminate $y^{2}$ from the system.

		$y^{2}$	$+$	$x^{2}$	$=$	$1$	$2$	$5$
$+$	$-$	$y^{2}$	$+$	$x^{2}$	$=$	$-$	$7$	$5$
			$2$	$x^{2}$	$=$		$5$	$0$

Step 4

Divide each term in $2 x^{2} = 50$ by $2$ and simplify.

Divide each term in $2 x^{2} = 50$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{50}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{50}{2}$

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

$x^{2} = \frac{50}{2}$

Simplify the right side.

Divide $50$ by $2$ .

$x^{2} = 25$

Step 5

Substitute the value found for $x^{2}$ into one of the original equations, then solve for $y^{2}$ .

Substitute the value found for $x^{2}$ into one of the original equations to solve for $y^{2}$ .

$y^{2} + 25 = 125$

Move all terms not containing $y^{2}$ to the right side of the equation.

Subtract $25$ from both sides of the equation.

$y^{2} = 125 - 25$

Subtract $25$ from $125$ .

$y^{2} = 100$

Step 6

This is the final solution to the independent system of equations.

$x^{2} = 25$

$y^{2} = 100$

Step 7

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

$x = \pm \sqrt{25}$

$y^{2} = 100$

Step 8

Simplify $\pm \sqrt{25}$ .

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt{5^{2}}$

$y^{2} = 100$

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

$x = \pm 5$

$y^{2} = 100$

$x = \pm 5$

$y^{2} = 100$

Step 9

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.

$x = 5$

$y^{2} = 100$

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

$x = - 5$

$y^{2} = 100$

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

$x = 5, - 5$

$y^{2} = 100$

$x = 5, - 5$

$y^{2} = 100$

Step 10

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

$y = \pm \sqrt{100}$

$x = 5, - 5$

Step 11

Simplify $\pm \sqrt{100}$ .

Rewrite $100$ as $10^{2}$ .

$y = \pm \sqrt{10^{2}}$

$x = 5, - 5$

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

$y = \pm 10$

$x = 5, - 5$

$y = \pm 10$

$x = 5, - 5$

Step 12

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.

$y = 10$

$x = 5, - 5$

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

$y = - 10$

$x = 5, - 5$

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

$y = 10, - 10$

$x = 5, - 5$

$y = 10, - 10$

$x = 5, - 5$

Step 13

The final result is the combination of all values of $x$ with all values of $y$ .

$(5, 10), (5, - 10), (- 5, 10), (- 5, - 10)$

Step 14

The result can be shown in multiple forms.

Point Form:

$(5, 10), (5, - 10), (- 5, 10), (- 5, - 10)$

Equation Form:

$x = 5, y = 10$

$x = 5, y = - 10$

$x = - 5, y = 10$

$x = - 5, y = - 10$

Step 15

Solve by Addition/Elimination x^2+y^2=125 , x^2-y^2=-75

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