Solve {l}{y-x=4}{y^2-5x^2=20}

$$\left. \begin{array} { l } { y - x = 4 } \\ { y ^ { 2 } - 5 x ^ { 2 } = 20 } \end{array} \right.$$

Solve for y, x

$$x=1$$

$$y=5$$

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.

$$y-x=4,-5x^{2}+y^{2}=20$$

Solve y-x=4 for y by isolating y on the left hand side of the equal sign.

$$y-x=4$$

Subtract -x from both sides of the equation.

$$y=x+4$$

Substitute x+4 for y in the other equation, -5x^{2}+y^{2}=20.

$$-5x^{2}+\left(x+4\right)^{2}=20$$

Square x+4.

$$-5x^{2}+x^{2}+8x+16=20$$

Add -5x^{2} to x^{2}.

$$-4x^{2}+8x+16=20$$

Subtract 20 from both sides of the equation.

$$-4x^{2}+8x-4=0$$

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

$$x=\frac{-8±\sqrt{8^{2}-4\left(-4\right)\left(-4\right)}}{2\left(-4\right)}$$

Square 1\times 4\times 1\times 2.

$$x=\frac{-8±\sqrt{64-4\left(-4\right)\left(-4\right)}}{2\left(-4\right)}$$

Multiply -4 times -5+1\times 1^{2}.

$$x=\frac{-8±\sqrt{64+16\left(-4\right)}}{2\left(-4\right)}$$

Multiply 16 times -4.

$$x=\frac{-8±\sqrt{64-64}}{2\left(-4\right)}$$

Add 64 to -64.

$$x=\frac{-8±\sqrt{0}}{2\left(-4\right)}$$

Take the square root of 0.

$$x=-\frac{8}{2\left(-4\right)}$$

Multiply 2 times -5+1\times 1^{2}.

$$x=-\frac{8}{-8}$$

Divide -8 by -8.

$$x=1$$

There are two solutions for x: 1 and 1. Substitute 1 for x in the equation y=x+4 to find the corresponding solution for y that satisfies both equations.

$$y=1+4$$

Add 1\times 1 to 4.

$$y=5$$

The system is now solved.

$$y=5,x=1\text{ or }y=5,x=1$$