Solve {l}{x^2+y^2=13}{x+y=10}

$$\left\{ \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = 13 } \\ { x + y = 10 } \end{array} \right.$$

Solve for x, y (complex solution)

$$x=\frac{\sqrt{74}i}{2}+5\approx 5+4.301162634i\text{, }y=-\frac{\sqrt{74}i}{2}+5\approx 5-4.301162634i$$

$$x=-\frac{\sqrt{74}i}{2}+5\approx 5-4.301162634i\text{, }y=\frac{\sqrt{74}i}{2}+5\approx 5+4.301162634i$$

Steps Using Substitution and Quadratic Formula

Solve x+y=10 for x by isolating x on the left hand side of the equal sign.

$$x+y=10$$

Subtract y from both sides of the equation.

$$x=-y+10$$

Substitute -y+10 for x in the other equation, y^{2}+x^{2}=13.

$$y^{2}+\left(-y+10\right)^{2}=13$$

Square -y+10.

$$y^{2}+y^{2}-20y+100=13$$

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

$$2y^{2}-20y+100=13$$

Subtract 13 from both sides of the equation.

$$2y^{2}-20y+87=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\left(-1\right)^{2} for a, 1\times 10\left(-1\right)\times 2 for b, and 87 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 2\times 87}}{2\times 2}$$

Square 1\times 10\left(-1\right)\times 2.

$$y=\frac{-\left(-20\right)±\sqrt{400-4\times 2\times 87}}{2\times 2}$$

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

$$y=\frac{-\left(-20\right)±\sqrt{400-8\times 87}}{2\times 2}$$

Multiply -8 times 87.

$$y=\frac{-\left(-20\right)±\sqrt{400-696}}{2\times 2}$$

Add 400 to -696.

$$y=\frac{-\left(-20\right)±\sqrt{-296}}{2\times 2}$$

Take the square root of -296.

$$y=\frac{-\left(-20\right)±2\sqrt{74}i}{2\times 2}$$

The opposite of 1\times 10\left(-1\right)\times 2 is 20.

$$y=\frac{20±2\sqrt{74}i}{2\times 2}$$

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

$$y=\frac{20±2\sqrt{74}i}{4}$$

Now solve the equation y=\frac{20±2\sqrt{74}i}{4} when ± is plus. Add 20 to 2i\sqrt{74}.

$$y=\frac{20+2\sqrt{74}i}{4}$$

Divide 20+2i\sqrt{74} by 4.

$$y=\frac{\sqrt{74}i}{2}+5$$

Now solve the equation y=\frac{20±2\sqrt{74}i}{4} when ± is minus. Subtract 2i\sqrt{74} from 20.

$$y=\frac{-2\sqrt{74}i+20}{4}$$

Divide 20-2i\sqrt{74} by 4.

$$y=-\frac{\sqrt{74}i}{2}+5$$

There are two solutions for y: 5+\frac{i\sqrt{74}}{2} and 5-\frac{i\sqrt{74}}{2}. Substitute 5+\frac{i\sqrt{74}}{2} for y in the equation x=-y+10 to find the corresponding solution for x that satisfies both equations.

$$x=-\left(\frac{\sqrt{74}i}{2}+5\right)+10$$

Now substitute 5-\frac{i\sqrt{74}}{2} for y in the equation x=-y+10 and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\left(-\frac{\sqrt{74}i}{2}+5\right)+10$$

The system is now solved.

$$x=-\left(\frac{\sqrt{74}i}{2}+5\right)+10,y=\frac{\sqrt{74}i}{2}+5\text{ or }x=-\left(-\frac{\sqrt{74}i}{2}+5\right)+10,y=-\frac{\sqrt{74}i}{2}+5$$