Solve {l}{x^2+y^2=3}{x+y=7}

$$\left. \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = 3 } \\ { x + y = 7 } \end{array} \right.$$

Solve for x, y (complex solution)

$$x=\frac{7+\sqrt{43}i}{2}\approx 3.5+3.278719262i\text{, }y=\frac{-\sqrt{43}i+7}{2}\approx 3.5-3.278719262i$$

$$x=\frac{-\sqrt{43}i+7}{2}\approx 3.5-3.278719262i\text{, }y=\frac{7+\sqrt{43}i}{2}\approx 3.5+3.278719262i$$

Steps Using Substitution and Quadratic Formula

Solve x+y=7 for x by isolating x on the left hand side of the equal sign.

$$x+y=7$$

Subtract y from both sides of the equation.

$$x=-y+7$$

Substitute -y+7 for x in the other equation, y^{2}+x^{2}=3.

$$y^{2}+\left(-y+7\right)^{2}=3$$

Square -y+7.

$$y^{2}+y^{2}-14y+49=3$$

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

$$2y^{2}-14y+49=3$$

Subtract 3 from both sides of the equation.

$$2y^{2}-14y+46=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\left(-1\right)^{2} for a, 1\times 7\left(-1\right)\times 2 for b, and 46 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 2\times 46}}{2\times 2}$$

Square 1\times 7\left(-1\right)\times 2.

$$y=\frac{-\left(-14\right)±\sqrt{196-4\times 2\times 46}}{2\times 2}$$

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

$$y=\frac{-\left(-14\right)±\sqrt{196-8\times 46}}{2\times 2}$$

Multiply -8 times 46.

$$y=\frac{-\left(-14\right)±\sqrt{196-368}}{2\times 2}$$

Add 196 to -368.

$$y=\frac{-\left(-14\right)±\sqrt{-172}}{2\times 2}$$

Take the square root of -172.

$$y=\frac{-\left(-14\right)±2\sqrt{43}i}{2\times 2}$$

The opposite of 1\times 7\left(-1\right)\times 2 is 14.

$$y=\frac{14±2\sqrt{43}i}{2\times 2}$$

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

$$y=\frac{14±2\sqrt{43}i}{4}$$

Now solve the equation y=\frac{14±2\sqrt{43}i}{4} when ± is plus. Add 14 to 2i\sqrt{43}.

$$y=\frac{14+2\sqrt{43}i}{4}$$

Divide 14+2i\sqrt{43} by 4.

$$y=\frac{7+\sqrt{43}i}{2}$$

Now solve the equation y=\frac{14±2\sqrt{43}i}{4} when ± is minus. Subtract 2i\sqrt{43} from 14.

$$y=\frac{-2\sqrt{43}i+14}{4}$$

Divide 14-2i\sqrt{43} by 4.

$$y=\frac{-\sqrt{43}i+7}{2}$$

There are two solutions for y: \frac{7+i\sqrt{43}}{2} and \frac{7-i\sqrt{43}}{2}. Substitute \frac{7+i\sqrt{43}}{2} for y in the equation x=-y+7 to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{7+\sqrt{43}i}{2}+7$$

Now substitute \frac{7-i\sqrt{43}}{2} for y in the equation x=-y+7 and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{-\sqrt{43}i+7}{2}+7$$

The system is now solved.

$$x=-\frac{7+\sqrt{43}i}{2}+7,y=\frac{7+\sqrt{43}i}{2}\text{ or }x=-\frac{-\sqrt{43}i+7}{2}+7,y=\frac{-\sqrt{43}i+7}{2}$$