Solve {l}{x^2+y^2=1}{y-x=5/13sqrt{2}}

$$\left\{ \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = 1 } \\ { y - x = \frac { 5 } { 13 } \sqrt { 2 } } \end{array} \right.$$

Solve for x, y

$$x=-\frac{17\sqrt{2}}{26}\approx -0.924678098\text{, }y=-\frac{7\sqrt{2}}{26}\approx -0.380749805$$

$$x=\frac{7\sqrt{2}}{26}\approx 0.380749805\text{, }y=\frac{17\sqrt{2}}{26}\approx 0.924678098$$

Steps Using Substitution and Quadratic Formula

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.

$$y-x=\frac{5\sqrt{2}}{13},x^{2}+y^{2}=1$$

Solve y-x=\frac{5\sqrt{2}}{13} for y by isolating y on the left hand side of the equal sign.

$$y-x=\frac{5\sqrt{2}}{13}$$

Subtract -x from both sides of the equation.

$$y=x+\frac{5\sqrt{2}}{13}$$

Substitute x+\frac{5\sqrt{2}}{13} for y in the other equation, x^{2}+y^{2}=1.

$$x^{2}+\left(x+\frac{5\sqrt{2}}{13}\right)^{2}=1$$

Square x+\frac{5\sqrt{2}}{13}.

$$x^{2}+x^{2}+2\times \frac{5\sqrt{2}}{13}x+\left(\frac{5\sqrt{2}}{13}\right)^{2}=1$$

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

$$2x^{2}+2\times \frac{5\sqrt{2}}{13}x+\left(\frac{5\sqrt{2}}{13}\right)^{2}=1$$

Subtract 1 from both sides of the equation.

$$2x^{2}+2\times \frac{5\sqrt{2}}{13}x+\left(\frac{5\sqrt{2}}{13}\right)^{2}-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\times 1^{2} for a, 1\times 1\times 2\times \frac{5\sqrt{2}}{13} for b, and -\frac{119}{169} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2\times \frac{5\sqrt{2}}{13}±\sqrt{\left(2\times \frac{5\sqrt{2}}{13}\right)^{2}-4\times 2\left(-\frac{119}{169}\right)}}{2\times 2}$$

Square 1\times 1\times 2\times \frac{5\sqrt{2}}{13}.

$$x=\frac{-2\times \frac{5\sqrt{2}}{13}±\sqrt{\frac{200}{169}-4\times 2\left(-\frac{119}{169}\right)}}{2\times 2}$$

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

$$x=\frac{-2\times \frac{5\sqrt{2}}{13}±\sqrt{\frac{200}{169}-8\left(-\frac{119}{169}\right)}}{2\times 2}$$

Multiply -8 times -\frac{119}{169}.

$$x=\frac{-2\times \frac{5\sqrt{2}}{13}±\sqrt{\frac{200+952}{169}}}{2\times 2}$$

Add \frac{200}{169} to \frac{952}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-2\times \frac{5\sqrt{2}}{13}±\sqrt{\frac{1152}{169}}}{2\times 2}$$

Take the square root of \frac{1152}{169}.

$$x=\frac{-2\times \frac{5\sqrt{2}}{13}±\frac{24\sqrt{2}}{13}}{2\times 2}$$

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

$$x=\frac{-\frac{10\sqrt{2}}{13}±\frac{24\sqrt{2}}{13}}{4}$$

Now solve the equation x=\frac{-\frac{10\sqrt{2}}{13}±\frac{24\sqrt{2}}{13}}{4} when ± is plus. Add -\frac{10\sqrt{2}}{13} to \frac{24\sqrt{2}}{13}.

$$x=\frac{14\sqrt{2}}{4\times 13}$$

Divide \frac{14\sqrt{2}}{13} by 4.

$$x=\frac{7\sqrt{2}}{26}$$

Now solve the equation x=\frac{-\frac{10\sqrt{2}}{13}±\frac{24\sqrt{2}}{13}}{4} when ± is minus. Subtract \frac{24\sqrt{2}}{13} from -\frac{10\sqrt{2}}{13}.

$$x=-\frac{\frac{34\sqrt{2}}{13}}{4}$$

Divide -\frac{34\sqrt{2}}{13} by 4.

$$x=-\frac{17\sqrt{2}}{26}$$

There are two solutions for x: \frac{7\sqrt{2}}{26} and -\frac{17\sqrt{2}}{26}. Substitute \frac{7\sqrt{2}}{26} for x in the equation y=x+\frac{5\sqrt{2}}{13} to find the corresponding solution for y that satisfies both equations.

$$y=\frac{7\sqrt{2}}{26}+\frac{5\sqrt{2}}{13}$$

Add 1\times \frac{7\sqrt{2}}{26} to \frac{5\sqrt{2}}{13}.

$$y=\frac{5\sqrt{2}}{13}+\frac{7\sqrt{2}}{26}$$

Now substitute -\frac{17\sqrt{2}}{26} for x in the equation y=x+\frac{5\sqrt{2}}{13} and solve to find the corresponding solution for y that satisfies both equations.

$$y=-\frac{17\sqrt{2}}{26}+\frac{5\sqrt{2}}{13}$$

Add 1\left(-\frac{17\sqrt{2}}{26}\right) to \frac{5\sqrt{2}}{13}.

$$y=\frac{5\sqrt{2}}{13}-\frac{17\sqrt{2}}{26}$$

The system is now solved.

$$y=\frac{5\sqrt{2}}{13}+\frac{7\sqrt{2}}{26},x=\frac{7\sqrt{2}}{26}\text{ or }y=\frac{5\sqrt{2}}{13}-\frac{17\sqrt{2}}{26},x=-\frac{17\sqrt{2}}{26}$$