Solve {l}{x^2+y^2=42}{y=x+13}

$$\left. \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = 42 } \\ { y = x + 13 } \end{array} \right.$$

Solve for x, y (complex solution)

$$x=\frac{-\sqrt{85}i-13}{2}\approx -6.5-4.609772229i\text{, }y=\frac{-\sqrt{85}i+13}{2}\approx 6.5-4.609772229i$$

$$x=\frac{-13+\sqrt{85}i}{2}\approx -6.5+4.609772229i\text{, }y=\frac{13+\sqrt{85}i}{2}\approx 6.5+4.609772229i$$

Steps Using Substitution and Quadratic Formula

Consider the second equation. Subtract x from both sides.

$$y-x=13$$

Subtract -x from both sides of the equation.

$$y=x+13$$

Substitute x+13 for y in the other equation, x^{2}+y^{2}=42.

$$x^{2}+\left(x+13\right)^{2}=42$$

Square x+13.

$$x^{2}+x^{2}+26x+169=42$$

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

$$2x^{2}+26x+169=42$$

Subtract 42 from both sides of the equation.

$$2x^{2}+26x+127=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\times 1^{2} for a, 1\times 13\times 1\times 2 for b, and 127 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-26±\sqrt{26^{2}-4\times 2\times 127}}{2\times 2}$$

Square 1\times 13\times 1\times 2.

$$x=\frac{-26±\sqrt{676-4\times 2\times 127}}{2\times 2}$$

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

$$x=\frac{-26±\sqrt{676-8\times 127}}{2\times 2}$$

Multiply -8 times 127.

$$x=\frac{-26±\sqrt{676-1016}}{2\times 2}$$

Add 676 to -1016.

$$x=\frac{-26±\sqrt{-340}}{2\times 2}$$

Take the square root of -340.

$$x=\frac{-26±2\sqrt{85}i}{2\times 2}$$

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

$$x=\frac{-26±2\sqrt{85}i}{4}$$

Now solve the equation x=\frac{-26±2\sqrt{85}i}{4} when ± is plus. Add -26 to 2i\sqrt{85}.

$$x=\frac{-26+2\sqrt{85}i}{4}$$

Divide -26+2i\sqrt{85} by 4.

$$x=\frac{-13+\sqrt{85}i}{2}$$

Now solve the equation x=\frac{-26±2\sqrt{85}i}{4} when ± is minus. Subtract 2i\sqrt{85} from -26.

$$x=\frac{-2\sqrt{85}i-26}{4}$$

Divide -26-2i\sqrt{85} by 4.

$$x=\frac{-\sqrt{85}i-13}{2}$$

There are two solutions for x: \frac{-13+i\sqrt{85}}{2} and \frac{-13-i\sqrt{85}}{2}. Substitute \frac{-13+i\sqrt{85}}{2} for x in the equation y=x+13 to find the corresponding solution for y that satisfies both equations.

$$y=\frac{-13+\sqrt{85}i}{2}+13$$

Now substitute \frac{-13-i\sqrt{85}}{2} for x in the equation y=x+13 and solve to find the corresponding solution for y that satisfies both equations.

$$y=\frac{-\sqrt{85}i-13}{2}+13$$

The system is now solved.

$$y=\frac{-13+\sqrt{85}i}{2}+13,x=\frac{-13+\sqrt{85}i}{2}\text{ or }y=\frac{-\sqrt{85}i-13}{2}+13,x=\frac{-\sqrt{85}i-13}{2}$$