Solve {l}{x^2+y^2=100}{x+y=-2}

$$\left. \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = 100 } \\ { x + y = - 2 } \end{array} \right.$$

Solve for x, y

$$x=6\text{, }y=-8$$

$$x=-8\text{, }y=6$$

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.

$$x+y=-2,y^{2}+x^{2}=100$$

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

$$x+y=-2$$

Subtract y from both sides of the equation.

$$x=-y-2$$

Substitute -y-2 for x in the other equation, y^{2}+x^{2}=100.

$$y^{2}+\left(-y-2\right)^{2}=100$$

Square -y-2.

$$y^{2}+y^{2}+4y+4=100$$

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

$$2y^{2}+4y+4=100$$

Subtract 100 from both sides of the equation.

$$2y^{2}+4y-96=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\left(-1\right)^{2} for a, 1\left(-2\right)\left(-1\right)\times 2 for b, and -96 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-4±\sqrt{4^{2}-4\times 2\left(-96\right)}}{2\times 2}$$

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

$$y=\frac{-4±\sqrt{16-4\times 2\left(-96\right)}}{2\times 2}$$

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

$$y=\frac{-4±\sqrt{16-8\left(-96\right)}}{2\times 2}$$

Multiply -8 times -96.

$$y=\frac{-4±\sqrt{16+768}}{2\times 2}$$

Add 16 to 768.

$$y=\frac{-4±\sqrt{784}}{2\times 2}$$

Take the square root of 784.

$$y=\frac{-4±28}{2\times 2}$$

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

$$y=\frac{-4±28}{4}$$

Now solve the equation y=\frac{-4±28}{4} when ± is plus. Add -4 to 28.

$$y=\frac{24}{4}$$

Divide 24 by 4.

$$y=6$$

Now solve the equation y=\frac{-4±28}{4} when ± is minus. Subtract 28 from -4.

$$y=-\frac{32}{4}$$

Divide -32 by 4.

$$y=-8$$

There are two solutions for y: 6 and -8. Substitute 6 for y in the equation x=-y-2 to find the corresponding solution for x that satisfies both equations.

$$x=-6-2$$

Add -6 to -2.

$$x=-8$$

Now substitute -8 for y in the equation x=-y-2 and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\left(-8\right)-2$$

Multiply -1 times -8.

$$x=8-2$$

Add -8\left(-1\right) to -2.

$$x=6$$

The system is now solved.

$$x=-8,y=6\text{ or }x=6,y=-8$$