Solve {l}{x^2+y^2=7}{x+y=20}

$$\left. \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = 7 } \\ { x + y = 20 } \end{array} \right.$$

Solve for x, y (complex solution)

$$x=\frac{\sqrt{386}i}{2}+10\approx 10+9.823441352i\text{, }y=-\frac{\sqrt{386}i}{2}+10\approx 10-9.823441352i$$

$$x=-\frac{\sqrt{386}i}{2}+10\approx 10-9.823441352i\text{, }y=\frac{\sqrt{386}i}{2}+10\approx 10+9.823441352i$$

Steps Using Substitution and Quadratic Formula

Solve x+y=20 for x by isolating x on the left hand side of the equal sign.

$$x+y=20$$

Subtract y from both sides of the equation.

$$x=-y+20$$

Substitute -y+20 for x in the other equation, y^{2}+x^{2}=7.

$$y^{2}+\left(-y+20\right)^{2}=7$$

Square -y+20.

$$y^{2}+y^{2}-40y+400=7$$

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

$$2y^{2}-40y+400=7$$

Subtract 7 from both sides of the equation.

$$2y^{2}-40y+393=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\left(-1\right)^{2} for a, 1\times 20\left(-1\right)\times 2 for b, and 393 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\times 2\times 393}}{2\times 2}$$

Square 1\times 20\left(-1\right)\times 2.

$$y=\frac{-\left(-40\right)±\sqrt{1600-4\times 2\times 393}}{2\times 2}$$

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

$$y=\frac{-\left(-40\right)±\sqrt{1600-8\times 393}}{2\times 2}$$

Multiply -8 times 393.

$$y=\frac{-\left(-40\right)±\sqrt{1600-3144}}{2\times 2}$$

Add 1600 to -3144.

$$y=\frac{-\left(-40\right)±\sqrt{-1544}}{2\times 2}$$

Take the square root of -1544.

$$y=\frac{-\left(-40\right)±2\sqrt{386}i}{2\times 2}$$

The opposite of 1\times 20\left(-1\right)\times 2 is 40.

$$y=\frac{40±2\sqrt{386}i}{2\times 2}$$

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

$$y=\frac{40±2\sqrt{386}i}{4}$$

Now solve the equation y=\frac{40±2\sqrt{386}i}{4} when ± is plus. Add 40 to 2i\sqrt{386}.

$$y=\frac{40+2\sqrt{386}i}{4}$$

Divide 40+2i\sqrt{386} by 4.

$$y=\frac{\sqrt{386}i}{2}+10$$

Now solve the equation y=\frac{40±2\sqrt{386}i}{4} when ± is minus. Subtract 2i\sqrt{386} from 40.

$$y=\frac{-2\sqrt{386}i+40}{4}$$

Divide 40-2i\sqrt{386} by 4.

$$y=-\frac{\sqrt{386}i}{2}+10$$

There are two solutions for y: 10+\frac{i\sqrt{386}}{2} and 10-\frac{i\sqrt{386}}{2}. Substitute 10+\frac{i\sqrt{386}}{2} for y in the equation x=-y+20 to find the corresponding solution for x that satisfies both equations.

$$x=-\left(\frac{\sqrt{386}i}{2}+10\right)+20$$

Now substitute 10-\frac{i\sqrt{386}}{2} for y in the equation x=-y+20 and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\left(-\frac{\sqrt{386}i}{2}+10\right)+20$$

The system is now solved.

$$x=-\left(\frac{\sqrt{386}i}{2}+10\right)+20,y=\frac{\sqrt{386}i}{2}+10\text{ or }x=-\left(-\frac{\sqrt{386}i}{2}+10\right)+20,y=-\frac{\sqrt{386}i}{2}+10$$