Solve {l}{3x^2-2y^2=25}{x+y=0}

$$\left. \begin{array}{l}{ 3 x ^ { 2 } - 2 y ^ { 2 } = 25 }\\{ x + y = 0 }\end{array} \right.$$

Solve for x, y

$$x=-5\text{, }y=5$$

$$x=5\text{, }y=-5$$

Steps Using Substitution and Quadratic Formula

To solve a pair of equations using substitution, first solve one of the equations for one of the variables. Then substitute the result for that variable in the other equation.

$$x+y=0,-2y^{2}+3x^{2}=25$$

Solve x+y=0 for x by isolating x on the left hand side of the equal sign.

$$x+y=0$$

Subtract y from both sides of the equation.

$$x=-y$$

Substitute -y for x in the other equation, -2y^{2}+3x^{2}=25.

$$-2y^{2}+3\left(-y\right)^{2}=25$$

Square -y.

$$-2y^{2}+3y^{2}=25$$

Add -2y^{2} to 3y^{2}.

$$y^{2}=25$$

Subtract 25 from both sides of the equation.

$$y^{2}-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2+3\left(-1\right)^{2} for a, 3\times 0\left(-1\right)\times 2 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{0±\sqrt{0^{2}-4\left(-25\right)}}{2}$$

Square 3\times 0\left(-1\right)\times 2.

$$y=\frac{0±\sqrt{-4\left(-25\right)}}{2}$$

Multiply -4 times -25.

$$y=\frac{0±\sqrt{100}}{2}$$

Take the square root of 100.

$$y=\frac{0±10}{2}$$

Now solve the equation y=\frac{0±10}{2} when ± is plus. Divide 10 by 2.

$$y=5$$

Now solve the equation y=\frac{0±10}{2} when ± is minus. Divide -10 by 2.

$$y=-5$$

There are two solutions for y: 5 and -5. Substitute 5 for y in the equation x=-y to find the corresponding solution for x that satisfies both equations.

$$x=-5$$

Now substitute -5 for y in the equation x=-y and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\left(-5\right)$$

Multiply -1 times -5.

$$x=5$$

The system is now solved.

$$x=-5,y=5\text{ or }x=5,y=-5$$