Solve {l}{x^2+y^2=20}{y-2x=0}

$$\left\{ \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = 20 } \\ { y - 2 x = 0 } \end{array} \right.$$

Solve for x, y

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

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

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.

$$y-2x=0,x^{2}+y^{2}=20$$

Solve y-2x=0 for y by isolating y on the left hand side of the equal sign.

$$y-2x=0$$

Subtract -2x from both sides of the equation.

$$y=2x$$

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

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

Square 2x.

$$x^{2}+4x^{2}=20$$

Add x^{2} to 4x^{2}.

$$5x^{2}=20$$

Subtract 20 from both sides of the equation.

$$5x^{2}-20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\times 2^{2} for a, 1\times 0\times 2\times 2 for b, and -20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 5\left(-20\right)}}{2\times 5}$$

Square 1\times 0\times 2\times 2.

$$x=\frac{0±\sqrt{-4\times 5\left(-20\right)}}{2\times 5}$$

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

$$x=\frac{0±\sqrt{-20\left(-20\right)}}{2\times 5}$$

Multiply -20 times -20.

$$x=\frac{0±\sqrt{400}}{2\times 5}$$

Take the square root of 400.

$$x=\frac{0±20}{2\times 5}$$

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

$$x=\frac{0±20}{10}$$

Now solve the equation x=\frac{0±20}{10} when ± is plus. Divide 20 by 10.

$$x=2$$

Now solve the equation x=\frac{0±20}{10} when ± is minus. Divide -20 by 10.

$$x=-2$$

There are two solutions for x: 2 and -2. Substitute 2 for x in the equation y=2x to find the corresponding solution for y that satisfies both equations.

$$y=2\times 2$$

Multiply 2 times 2.

$$y=4$$

Now substitute -2 for x in the equation y=2x and solve to find the corresponding solution for y that satisfies both equations.

$$y=2\left(-2\right)$$

Multiply 2 times -2.

$$y=-4$$

The system is now solved.

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