Solve {l}{x^2+y^2=1}{y+4x=7}

$$\left. \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = 1 } \\ { y + 4 x = 7 } \end{array} \right.$$

Solve for x, y (complex solution)

$$x=\frac{28+4\sqrt{2}i}{17}\approx 1.647058824+0.332756132i\text{, }y=\frac{-16\sqrt{2}i+7}{17}\approx 0.411764706-1.331024529i$$

$$x=\frac{-4\sqrt{2}i+28}{17}\approx 1.647058824-0.332756132i\text{, }y=\frac{7+16\sqrt{2}i}{17}\approx 0.411764706+1.331024529i$$

Steps Using Substitution and Quadratic Formula

Solve y+4x=7 for y by isolating y on the left hand side of the equal sign.

$$y+4x=7$$

Subtract 4x from both sides of the equation.

$$y=-4x+7$$

Substitute -4x+7 for y in the other equation, x^{2}+y^{2}=1.

$$x^{2}+\left(-4x+7\right)^{2}=1$$

Square -4x+7.

$$x^{2}+16x^{2}-56x+49=1$$

Add x^{2} to 16x^{2}.

$$17x^{2}-56x+49=1$$

Subtract 1 from both sides of the equation.

$$17x^{2}-56x+48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\left(-4\right)^{2} for a, 1\times 7\left(-4\right)\times 2 for b, and 48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-56\right)±\sqrt{\left(-56\right)^{2}-4\times 17\times 48}}{2\times 17}$$

Square 1\times 7\left(-4\right)\times 2.

$$x=\frac{-\left(-56\right)±\sqrt{3136-4\times 17\times 48}}{2\times 17}$$

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

$$x=\frac{-\left(-56\right)±\sqrt{3136-68\times 48}}{2\times 17}$$

Multiply -68 times 48.

$$x=\frac{-\left(-56\right)±\sqrt{3136-3264}}{2\times 17}$$

Add 3136 to -3264.

$$x=\frac{-\left(-56\right)±\sqrt{-128}}{2\times 17}$$

Take the square root of -128.

$$x=\frac{-\left(-56\right)±8\sqrt{2}i}{2\times 17}$$

The opposite of 1\times 7\left(-4\right)\times 2 is 56.

$$x=\frac{56±8\sqrt{2}i}{2\times 17}$$

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

$$x=\frac{56±8\sqrt{2}i}{34}$$

Now solve the equation x=\frac{56±8\sqrt{2}i}{34} when ± is plus. Add 56 to 8i\sqrt{2}.

$$x=\frac{56+2^{\frac{7}{2}}i}{34}$$

Divide 56+i\times 2^{\frac{7}{2}} by 34.

$$x=\frac{28+4\sqrt{2}i}{17}$$

Now solve the equation x=\frac{56±8\sqrt{2}i}{34} when ± is minus. Subtract 8i\sqrt{2} from 56.

$$x=\frac{-2^{\frac{7}{2}}i+56}{34}$$

Divide 56-i\times 2^{\frac{7}{2}} by 34.

$$x=\frac{-4\sqrt{2}i+28}{17}$$

There are two solutions for x: \frac{28+4i\sqrt{2}}{17} and \frac{28-4i\sqrt{2}}{17}. Substitute \frac{28+4i\sqrt{2}}{17} for x in the equation y=-4x+7 to find the corresponding solution for y that satisfies both equations.

$$y=-4\times \frac{28+4\sqrt{2}i}{17}+7$$

Now substitute \frac{28-4i\sqrt{2}}{17} for x in the equation y=-4x+7 and solve to find the corresponding solution for y that satisfies both equations.

$$y=-4\times \frac{-4\sqrt{2}i+28}{17}+7$$

The system is now solved.

$$y=-4\times \frac{28+4\sqrt{2}i}{17}+7,x=\frac{28+4\sqrt{2}i}{17}\text{ or }y=-4\times \frac{-4\sqrt{2}i+28}{17}+7,x=\frac{-4\sqrt{2}i+28}{17}$$