Solve {l}{8y=-10x}{y^2=2x^2-7}

$$\left. \begin{array} { l } { 8 y = - 10 x } \\ { y ^ { 2 } = 2 x ^ { 2 } - 7 } \end{array} \right.$$

Solve for y, x

$$x=-4\text{, }y=5$$

$$x=4\text{, }y=-5$$

Steps Using Substitution and Quadratic Formula

Consider the first equation. Add 10x to both sides.

$$8y+10x=0$$

Consider the second equation. Subtract 2x^{2} from both sides.

$$y^{2}-2x^{2}=-7$$

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.

$$8y+10x=0,-2x^{2}+y^{2}=-7$$

Solve 8y+10x=0 for y by isolating y on the left hand side of the equal sign.

$$8y+10x=0$$

Subtract 10x from both sides of the equation.

$$8y=-10x$$

Divide both sides by 8.

$$y=-\frac{5}{4}x$$

Substitute -\frac{5}{4}x for y in the other equation, -2x^{2}+y^{2}=-7.

$$-2x^{2}+\left(-\frac{5}{4}x\right)^{2}=-7$$

Square -\frac{5}{4}x.

$$-2x^{2}+\frac{25}{16}x^{2}=-7$$

Add -2x^{2} to \frac{25}{16}x^{2}.

$$-\frac{7}{16}x^{2}=-7$$

Add 7 to both sides of the equation.

$$-\frac{7}{16}x^{2}+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2+1\left(-\frac{5}{4}\right)^{2} for a, 1\times 0\left(-\frac{5}{4}\right)\times 2 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-\frac{7}{16}\right)\times 7}}{2\left(-\frac{7}{16}\right)}$$

Square 1\times 0\left(-\frac{5}{4}\right)\times 2.

$$x=\frac{0±\sqrt{-4\left(-\frac{7}{16}\right)\times 7}}{2\left(-\frac{7}{16}\right)}$$

Multiply -4 times -2+1\left(-\frac{5}{4}\right)^{2}.

$$x=\frac{0±\sqrt{\frac{7}{4}\times 7}}{2\left(-\frac{7}{16}\right)}$$

Multiply \frac{7}{4} times 7.

$$x=\frac{0±\sqrt{\frac{49}{4}}}{2\left(-\frac{7}{16}\right)}$$

Take the square root of \frac{49}{4}.

$$x=\frac{0±\frac{7}{2}}{2\left(-\frac{7}{16}\right)}$$

Multiply 2 times -2+1\left(-\frac{5}{4}\right)^{2}.

$$x=\frac{0±\frac{7}{2}}{-\frac{7}{8}}$$

Now solve the equation x=\frac{0±\frac{7}{2}}{-\frac{7}{8}} when ± is plus.

$$x=-4$$

Now solve the equation x=\frac{0±\frac{7}{2}}{-\frac{7}{8}} when ± is minus.

$$x=4$$

There are two solutions for x: -4 and 4. Substitute -4 for x in the equation y=-\frac{5}{4}x to find the corresponding solution for y that satisfies both equations.

$$y=-\frac{5}{4}\left(-4\right)$$

Multiply -\frac{5}{4} times -4.

$$y=5$$

Now substitute 4 for x in the equation y=-\frac{5}{4}x and solve to find the corresponding solution for y that satisfies both equations.

$$y=-\frac{5}{4}\times 4$$

Multiply -\frac{5}{4} times 4.

$$y=-5$$

The system is now solved.

$$y=5,x=-4\text{ or }y=-5,x=4$$