Solve {l}{x/y=-8}{x^2-y^2=40}

$$\left\{ \begin{array} { l } { \frac { x } { y } = - 8 } \\ { x ^ { 2 } - y ^ { 2 } = 40 } \end{array} \right.$$

Solve for x, y

$$x=\frac{16\sqrt{70}}{21}\approx 6.374552583\text{, }y=-\frac{2\sqrt{70}}{21}\approx -0.796819073$$

$$x=-\frac{16\sqrt{70}}{21}\approx -6.374552583\text{, }y=\frac{2\sqrt{70}}{21}\approx 0.796819073$$

Steps Using Substitution and Quadratic Formula

Consider the first equation. Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by y.

$$x=-8y$$

Substitute -8y for x in the other equation, -y^{2}+x^{2}=40.

$$-y^{2}+\left(-8y\right)^{2}=40$$

Square -8y.

$$-y^{2}+64y^{2}=40$$

Add -y^{2} to 64y^{2}.

$$63y^{2}=40$$

Subtract 40 from both sides of the equation.

$$63y^{2}-40=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1+1\left(-8\right)^{2} for a, 1\times 0\left(-8\right)\times 2 for b, and -40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{0±\sqrt{0^{2}-4\times 63\left(-40\right)}}{2\times 63}$$

Square 1\times 0\left(-8\right)\times 2.

$$y=\frac{0±\sqrt{-4\times 63\left(-40\right)}}{2\times 63}$$

Multiply -4 times -1+1\left(-8\right)^{2}.

$$y=\frac{0±\sqrt{-252\left(-40\right)}}{2\times 63}$$

Multiply -252 times -40.

$$y=\frac{0±\sqrt{10080}}{2\times 63}$$

Take the square root of 10080.

$$y=\frac{0±12\sqrt{70}}{2\times 63}$$

Multiply 2 times -1+1\left(-8\right)^{2}.

$$y=\frac{0±12\sqrt{70}}{126}$$

Now solve the equation y=\frac{0±12\sqrt{70}}{126} when ± is plus.

$$y=\frac{2\sqrt{70}}{21}$$

Now solve the equation y=\frac{0±12\sqrt{70}}{126} when ± is minus.

$$y=-\frac{2\sqrt{70}}{21}$$

There are two solutions for y: \frac{2\sqrt{70}}{21} and -\frac{2\sqrt{70}}{21}. Substitute \frac{2\sqrt{70}}{21} for y in the equation x=-8y to find the corresponding solution for x that satisfies both equations.

$$x=-8\times \frac{2\sqrt{70}}{21}$$

Now substitute -\frac{2\sqrt{70}}{21} for y in the equation x=-8y and solve to find the corresponding solution for x that satisfies both equations.

$$x=-8\left(-\frac{2\sqrt{70}}{21}\right)$$

The system is now solved.

$$x=-8\times \frac{2\sqrt{70}}{21},y=\frac{2\sqrt{70}}{21}\text{ or }x=-8\left(-\frac{2\sqrt{70}}{21}\right),y=-\frac{2\sqrt{70}}{21}$$