Solve {l}{x^2+y^2=2}{x-y=5}

$$\left. \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = 2 } \\ { x - y = 5 } \end{array} \right.$$

Solve for x, y (complex solution)

$$x=\frac{-\sqrt{21}i+5}{2}\approx 2.5-2.291287847i\text{, }y=\frac{-\sqrt{21}i-5}{2}\approx -2.5-2.291287847i$$

$$x=\frac{5+\sqrt{21}i}{2}\approx 2.5+2.291287847i\text{, }y=\frac{-5+\sqrt{21}i}{2}\approx -2.5+2.291287847i$$

Steps Using Substitution and Quadratic Formula

Solve x-y=5 for x by isolating x on the left hand side of the equal sign.

$$x-y=5$$

Subtract -y from both sides of the equation.

$$x=y+5$$

Substitute y+5 for x in the other equation, y^{2}+x^{2}=2.

$$y^{2}+\left(y+5\right)^{2}=2$$

Square y+5.

$$y^{2}+y^{2}+10y+25=2$$

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

$$2y^{2}+10y+25=2$$

Subtract 2 from both sides of the equation.

$$2y^{2}+10y+23=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\times 1^{2} for a, 1\times 5\times 1\times 2 for b, and 23 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-10±\sqrt{10^{2}-4\times 2\times 23}}{2\times 2}$$

Square 1\times 5\times 1\times 2.

$$y=\frac{-10±\sqrt{100-4\times 2\times 23}}{2\times 2}$$

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

$$y=\frac{-10±\sqrt{100-8\times 23}}{2\times 2}$$

Multiply -8 times 23.

$$y=\frac{-10±\sqrt{100-184}}{2\times 2}$$

Add 100 to -184.

$$y=\frac{-10±\sqrt{-84}}{2\times 2}$$

Take the square root of -84.

$$y=\frac{-10±2\sqrt{21}i}{2\times 2}$$

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

$$y=\frac{-10±2\sqrt{21}i}{4}$$

Now solve the equation y=\frac{-10±2\sqrt{21}i}{4} when ± is plus. Add -10 to 2i\sqrt{21}.

$$y=\frac{-10+2\sqrt{21}i}{4}$$

Divide -10+2i\sqrt{21} by 4.

$$y=\frac{-5+\sqrt{21}i}{2}$$

Now solve the equation y=\frac{-10±2\sqrt{21}i}{4} when ± is minus. Subtract 2i\sqrt{21} from -10.

$$y=\frac{-2\sqrt{21}i-10}{4}$$

Divide -10-2i\sqrt{21} by 4.

$$y=\frac{-\sqrt{21}i-5}{2}$$

There are two solutions for y: \frac{-5+i\sqrt{21}}{2} and \frac{-5-i\sqrt{21}}{2}. Substitute \frac{-5+i\sqrt{21}}{2} for y in the equation x=y+5 to find the corresponding solution for x that satisfies both equations.

$$x=\frac{-5+\sqrt{21}i}{2}+5$$

Now substitute \frac{-5-i\sqrt{21}}{2} for y in the equation x=y+5 and solve to find the corresponding solution for x that satisfies both equations.

$$x=\frac{-\sqrt{21}i-5}{2}+5$$

The system is now solved.

$$x=\frac{-5+\sqrt{21}i}{2}+5,y=\frac{-5+\sqrt{21}i}{2}\text{ or }x=\frac{-\sqrt{21}i-5}{2}+5,y=\frac{-\sqrt{21}i-5}{2}$$