Solve {l}{x^2+y^2=36}{x+2y=16}

$$\left. \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = 36 } \\ { x + 2 y = 16 } \end{array} \right.$$

Solve for x, y (complex solution)

$$x=\frac{16+4\sqrt{19}i}{5}\approx 3.2+3.487119155i\text{, }y=\frac{-2\sqrt{19}i+32}{5}\approx 6.4-1.743559577i$$

$$x=\frac{-4\sqrt{19}i+16}{5}\approx 3.2-3.487119155i\text{, }y=\frac{32+2\sqrt{19}i}{5}\approx 6.4+1.743559577i$$

Steps Using Substitution and Quadratic Formula

Solve x+2y=16 for x by isolating x on the left hand side of the equal sign.

$$x+2y=16$$

Subtract 2y from both sides of the equation.

$$x=-2y+16$$

Substitute -2y+16 for x in the other equation, y^{2}+x^{2}=36.

$$y^{2}+\left(-2y+16\right)^{2}=36$$

Square -2y+16.

$$y^{2}+4y^{2}-64y+256=36$$

Add y^{2} to 4y^{2}.

$$5y^{2}-64y+256=36$$

Subtract 36 from both sides of the equation.

$$5y^{2}-64y+220=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\left(-2\right)^{2} for a, 1\times 16\left(-2\right)\times 2 for b, and 220 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-64\right)±\sqrt{\left(-64\right)^{2}-4\times 5\times 220}}{2\times 5}$$

Square 1\times 16\left(-2\right)\times 2.

$$y=\frac{-\left(-64\right)±\sqrt{4096-4\times 5\times 220}}{2\times 5}$$

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

$$y=\frac{-\left(-64\right)±\sqrt{4096-20\times 220}}{2\times 5}$$

Multiply -20 times 220.

$$y=\frac{-\left(-64\right)±\sqrt{4096-4400}}{2\times 5}$$

Add 4096 to -4400.

$$y=\frac{-\left(-64\right)±\sqrt{-304}}{2\times 5}$$

Take the square root of -304.

$$y=\frac{-\left(-64\right)±4\sqrt{19}i}{2\times 5}$$

The opposite of 1\times 16\left(-2\right)\times 2 is 64.

$$y=\frac{64±4\sqrt{19}i}{2\times 5}$$

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

$$y=\frac{64±4\sqrt{19}i}{10}$$

Now solve the equation y=\frac{64±4\sqrt{19}i}{10} when ± is plus. Add 64 to 4i\sqrt{19}.

$$y=\frac{64+4\sqrt{19}i}{10}$$

Divide 64+4i\sqrt{19} by 10.

$$y=\frac{32+2\sqrt{19}i}{5}$$

Now solve the equation y=\frac{64±4\sqrt{19}i}{10} when ± is minus. Subtract 4i\sqrt{19} from 64.

$$y=\frac{-4\sqrt{19}i+64}{10}$$

Divide 64-4i\sqrt{19} by 10.

$$y=\frac{-2\sqrt{19}i+32}{5}$$

There are two solutions for y: \frac{32+2i\sqrt{19}}{5} and \frac{32-2i\sqrt{19}}{5}. Substitute \frac{32+2i\sqrt{19}}{5} for y in the equation x=-2y+16 to find the corresponding solution for x that satisfies both equations.

$$x=-2\times \frac{32+2\sqrt{19}i}{5}+16$$

Now substitute \frac{32-2i\sqrt{19}}{5} for y in the equation x=-2y+16 and solve to find the corresponding solution for x that satisfies both equations.

$$x=-2\times \frac{-2\sqrt{19}i+32}{5}+16$$

The system is now solved.

$$x=-2\times \frac{32+2\sqrt{19}i}{5}+16,y=\frac{32+2\sqrt{19}i}{5}\text{ or }x=-2\times \frac{-2\sqrt{19}i+32}{5}+16,y=\frac{-2\sqrt{19}i+32}{5}$$