Solve {l}{x^2+y^2=1}{x+y=4}

$$\left. \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = 1 } \\ { x + y = 4 } \end{array} \right.$$

Solve for x, y (complex solution)

$$x=\frac{\sqrt{14}i}{2}+2\approx 2+1.870828693i\text{, }y=-\frac{\sqrt{14}i}{2}+2\approx 2-1.870828693i$$

$$x=-\frac{\sqrt{14}i}{2}+2\approx 2-1.870828693i\text{, }y=\frac{\sqrt{14}i}{2}+2\approx 2+1.870828693i$$

Steps Using Substitution and Quadratic Formula

Solve x+y=4 for x by isolating x on the left hand side of the equal sign.

$$x+y=4$$

Subtract y from both sides of the equation.

$$x=-y+4$$

Substitute -y+4 for x in the other equation, y^{2}+x^{2}=1.

$$y^{2}+\left(-y+4\right)^{2}=1$$

Square -y+4.

$$y^{2}+y^{2}-8y+16=1$$

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

$$2y^{2}-8y+16=1$$

Subtract 1 from both sides of the equation.

$$2y^{2}-8y+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\left(-1\right)^{2} for a, 1\times 4\left(-1\right)\times 2 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 2\times 15}}{2\times 2}$$

Square 1\times 4\left(-1\right)\times 2.

$$y=\frac{-\left(-8\right)±\sqrt{64-4\times 2\times 15}}{2\times 2}$$

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

$$y=\frac{-\left(-8\right)±\sqrt{64-8\times 15}}{2\times 2}$$

Multiply -8 times 15.

$$y=\frac{-\left(-8\right)±\sqrt{64-120}}{2\times 2}$$

Add 64 to -120.

$$y=\frac{-\left(-8\right)±\sqrt{-56}}{2\times 2}$$

Take the square root of -56.

$$y=\frac{-\left(-8\right)±2\sqrt{14}i}{2\times 2}$$

The opposite of 1\times 4\left(-1\right)\times 2 is 8.

$$y=\frac{8±2\sqrt{14}i}{2\times 2}$$

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

$$y=\frac{8±2\sqrt{14}i}{4}$$

Now solve the equation y=\frac{8±2\sqrt{14}i}{4} when ± is plus. Add 8 to 2i\sqrt{14}.

$$y=\frac{8+2\sqrt{14}i}{4}$$

Divide 8+2i\sqrt{14} by 4.

$$y=\frac{\sqrt{14}i}{2}+2$$

Now solve the equation y=\frac{8±2\sqrt{14}i}{4} when ± is minus. Subtract 2i\sqrt{14} from 8.

$$y=\frac{-2\sqrt{14}i+8}{4}$$

Divide 8-2i\sqrt{14} by 4.

$$y=-\frac{\sqrt{14}i}{2}+2$$

There are two solutions for y: 2+\frac{i\sqrt{14}}{2} and 2-\frac{i\sqrt{14}}{2}. Substitute 2+\frac{i\sqrt{14}}{2} for y in the equation x=-y+4 to find the corresponding solution for x that satisfies both equations.

$$x=-\left(\frac{\sqrt{14}i}{2}+2\right)+4$$

Now substitute 2-\frac{i\sqrt{14}}{2} for y in the equation x=-y+4 and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\left(-\frac{\sqrt{14}i}{2}+2\right)+4$$

The system is now solved.

$$x=-\left(\frac{\sqrt{14}i}{2}+2\right)+4,y=\frac{\sqrt{14}i}{2}+2\text{ or }x=-\left(-\frac{\sqrt{14}i}{2}+2\right)+4,y=-\frac{\sqrt{14}i}{2}+2$$