Solve {l}{y-2=4x-16}{x^2+y^2=22}

$$\left\{ \begin{array} { l } { y - 2 = 4 x - 16 } \\ { x ^ { 2 } + y ^ { 2 } = 22 } \end{array} \right.$$

Solve for y, x

$$x=\frac{56-\sqrt{178}}{17}\approx 2.509313879\text{, }y=\frac{-4\sqrt{178}-14}{17}\approx -3.962744486$$

$$x=\frac{\sqrt{178}+56}{17}\approx 4.078921416\text{, }y=\frac{4\sqrt{178}-14}{17}\approx 2.315685662$$

Steps Using Substitution and Quadratic Formula

Consider the first equation. Subtract 4x from both sides.

$$y-2-4x=-16$$

Add 2 to both sides.

$$y-4x=-16+2$$

Add -16 and 2 to get -14.

$$y-4x=-14$$

To solve a pair of equations using substitution, first solve one of the equations for one of the variables. Then substitute the result for that variable in the other equation.

$$y-4x=-14,x^{2}+y^{2}=22$$

Solve y-4x=-14 for y by isolating y on the left hand side of the equal sign.

$$y-4x=-14$$

Subtract -4x from both sides of the equation.

$$y=4x-14$$

Substitute 4x-14 for y in the other equation, x^{2}+y^{2}=22.

$$x^{2}+\left(4x-14\right)^{2}=22$$

Square 4x-14.

$$x^{2}+16x^{2}-112x+196=22$$

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

$$17x^{2}-112x+196=22$$

Subtract 22 from both sides of the equation.

$$17x^{2}-112x+174=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\times 4^{2} for a, 1\left(-14\right)\times 2\times 4 for b, and 174 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-112\right)±\sqrt{\left(-112\right)^{2}-4\times 17\times 174}}{2\times 17}$$

Square 1\left(-14\right)\times 2\times 4.

$$x=\frac{-\left(-112\right)±\sqrt{12544-4\times 17\times 174}}{2\times 17}$$

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

$$x=\frac{-\left(-112\right)±\sqrt{12544-68\times 174}}{2\times 17}$$

Multiply -68 times 174.

$$x=\frac{-\left(-112\right)±\sqrt{12544-11832}}{2\times 17}$$

Add 12544 to -11832.

$$x=\frac{-\left(-112\right)±\sqrt{712}}{2\times 17}$$

Take the square root of 712.

$$x=\frac{-\left(-112\right)±2\sqrt{178}}{2\times 17}$$

The opposite of 1\left(-14\right)\times 2\times 4 is 112.

$$x=\frac{112±2\sqrt{178}}{2\times 17}$$

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

$$x=\frac{112±2\sqrt{178}}{34}$$

Now solve the equation x=\frac{112±2\sqrt{178}}{34} when ± is plus. Add 112 to 2\sqrt{178}.

$$x=\frac{2\sqrt{178}+112}{34}$$

Divide 112+2\sqrt{178} by 34.

$$x=\frac{\sqrt{178}+56}{17}$$

Now solve the equation x=\frac{112±2\sqrt{178}}{34} when ± is minus. Subtract 2\sqrt{178} from 112.

$$x=\frac{112-2\sqrt{178}}{34}$$

Divide 112-2\sqrt{178} by 34.

$$x=\frac{56-\sqrt{178}}{17}$$

There are two solutions for x: \frac{56+\sqrt{178}}{17} and \frac{56-\sqrt{178}}{17}. Substitute \frac{56+\sqrt{178}}{17} for x in the equation y=4x-14 to find the corresponding solution for y that satisfies both equations.

$$y=4\times \frac{\sqrt{178}+56}{17}-14$$

Now substitute \frac{56-\sqrt{178}}{17} for x in the equation y=4x-14 and solve to find the corresponding solution for y that satisfies both equations.

$$y=4\times \frac{56-\sqrt{178}}{17}-14$$

The system is now solved.

$$y=4\times \frac{\sqrt{178}+56}{17}-14,x=\frac{\sqrt{178}+56}{17}\text{ or }y=4\times \frac{56-\sqrt{178}}{17}-14,x=\frac{56-\sqrt{178}}{17}$$