Solve {l}{y^2-2x^2=8}{x+y=6}

$$\left\{ \begin{array} { l } { y ^ { 2 } - 2 x ^ { 2 } = 8 } \\ { x + y = 6 } \end{array} \right.$$

Solve for y, x

$$x=2\text{, }y=4$$

$$x=-14\text{, }y=20$$

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.

$$x+y=6,y^{2}-2x^{2}=8$$

Solve x+y=6 for x by isolating x on the left hand side of the equal sign.

$$x+y=6$$

Subtract y from both sides of the equation.

$$x=-y+6$$

Substitute -y+6 for x in the other equation, y^{2}-2x^{2}=8.

$$y^{2}-2\left(-y+6\right)^{2}=8$$

Square -y+6.

$$y^{2}-2\left(y^{2}-12y+36\right)=8$$

Multiply -2 times y^{2}-12y+36.

$$y^{2}-2y^{2}+24y-72=8$$

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

$$-y^{2}+24y-72=8$$

Subtract 8 from both sides of the equation.

$$-y^{2}+24y-80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1-2\left(-1\right)^{2} for a, -2\times 6\left(-1\right)\times 2 for b, and -80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-24±\sqrt{24^{2}-4\left(-1\right)\left(-80\right)}}{2\left(-1\right)}$$

Square -2\times 6\left(-1\right)\times 2.

$$y=\frac{-24±\sqrt{576-4\left(-1\right)\left(-80\right)}}{2\left(-1\right)}$$

Multiply -4 times 1-2\left(-1\right)^{2}.

$$y=\frac{-24±\sqrt{576+4\left(-80\right)}}{2\left(-1\right)}$$

Multiply 4 times -80.

$$y=\frac{-24±\sqrt{576-320}}{2\left(-1\right)}$$

Add 576 to -320.

$$y=\frac{-24±\sqrt{256}}{2\left(-1\right)}$$

Take the square root of 256.

$$y=\frac{-24±16}{2\left(-1\right)}$$

Multiply 2 times 1-2\left(-1\right)^{2}.

$$y=\frac{-24±16}{-2}$$

Now solve the equation y=\frac{-24±16}{-2} when ± is plus. Add -24 to 16.

$$y=-\frac{8}{-2}$$

Divide -8 by -2.

$$y=4$$

Now solve the equation y=\frac{-24±16}{-2} when ± is minus. Subtract 16 from -24.

$$y=-\frac{40}{-2}$$

Divide -40 by -2.

$$y=20$$

There are two solutions for y: 4 and 20. Substitute 4 for y in the equation x=-y+6 to find the corresponding solution for x that satisfies both equations.

$$x=-4+6$$

Add -4 to 6.

$$x=2$$

Now substitute 20 for y in the equation x=-y+6 and solve to find the corresponding solution for x that satisfies both equations.

$$x=-20+6$$

Add -20 to 6.

$$x=-14$$

The system is now solved.

$$x=2,y=4\text{ or }x=-14,y=20$$