Solve {l}{m-n=3/5sqrt{2}}{m^2+n^2=1}

$$\left\{ \begin{array} { l } { m - n = \frac { 3 } { 5 } \sqrt { 2 } } \\ { m ^ { 2 } + n ^ { 2 } = 1 } \end{array} \right.$$

Solve for m, n

$$m=-\frac{\sqrt{2}}{10}\approx -0.141421356\text{, }n=-\frac{7\sqrt{2}}{10}\approx -0.989949494$$

$$m=\frac{7\sqrt{2}}{10}\approx 0.989949494\text{, }n=\frac{\sqrt{2}}{10}\approx 0.141421356$$

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.

$$m-n=\frac{3\sqrt{2}}{5},n^{2}+m^{2}=1$$

Solve m-n=\frac{3\sqrt{2}}{5} for m by isolating m on the left hand side of the equal sign.

$$m-n=\frac{3\sqrt{2}}{5}$$

Subtract -n from both sides of the equation.

$$m=n+\frac{3\sqrt{2}}{5}$$

Substitute n+\frac{3\sqrt{2}}{5} for m in the other equation, n^{2}+m^{2}=1.

$$n^{2}+\left(n+\frac{3\sqrt{2}}{5}\right)^{2}=1$$

Square n+\frac{3\sqrt{2}}{5}.

$$n^{2}+n^{2}+2\times \frac{3\sqrt{2}}{5}n+\left(\frac{3\sqrt{2}}{5}\right)^{2}=1$$

Add n^{2} to n^{2}.

$$2n^{2}+2\times \frac{3\sqrt{2}}{5}n+\left(\frac{3\sqrt{2}}{5}\right)^{2}=1$$

Subtract 1 from both sides of the equation.

$$2n^{2}+2\times \frac{3\sqrt{2}}{5}n+\left(\frac{3\sqrt{2}}{5}\right)^{2}-1=0$$

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{3\sqrt{2}}{5} for b, and -\frac{7}{25} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-2\times \frac{3\sqrt{2}}{5}±\sqrt{\left(2\times \frac{3\sqrt{2}}{5}\right)^{2}-4\times 2\left(-\frac{7}{25}\right)}}{2\times 2}$$

Square 1\times 1\times 2\times \frac{3\sqrt{2}}{5}.

$$n=\frac{-2\times \frac{3\sqrt{2}}{5}±\sqrt{\frac{72}{25}-4\times 2\left(-\frac{7}{25}\right)}}{2\times 2}$$

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

$$n=\frac{-2\times \frac{3\sqrt{2}}{5}±\sqrt{\frac{72}{25}-8\left(-\frac{7}{25}\right)}}{2\times 2}$$

Multiply -8 times -\frac{7}{25}.

$$n=\frac{-2\times \frac{3\sqrt{2}}{5}±\sqrt{\frac{72+56}{25}}}{2\times 2}$$

Add \frac{72}{25} to \frac{56}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n=\frac{-2\times \frac{3\sqrt{2}}{5}±\sqrt{\frac{128}{25}}}{2\times 2}$$

Take the square root of \frac{128}{25}.

$$n=\frac{-2\times \frac{3\sqrt{2}}{5}±\frac{8\sqrt{2}}{5}}{2\times 2}$$

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

$$n=\frac{-\frac{6\sqrt{2}}{5}±\frac{8\sqrt{2}}{5}}{4}$$

Now solve the equation n=\frac{-\frac{6\sqrt{2}}{5}±\frac{8\sqrt{2}}{5}}{4} when ± is plus. Add -\frac{6\sqrt{2}}{5} to \frac{8\sqrt{2}}{5}.

$$n=\frac{2\sqrt{2}}{4\times 5}$$

Divide \frac{2\sqrt{2}}{5} by 4.

$$n=\frac{\sqrt{2}}{10}$$

Now solve the equation n=\frac{-\frac{6\sqrt{2}}{5}±\frac{8\sqrt{2}}{5}}{4} when ± is minus. Subtract \frac{8\sqrt{2}}{5} from -\frac{6\sqrt{2}}{5}.

$$n=-\frac{\frac{14\sqrt{2}}{5}}{4}$$

Divide -\frac{14\sqrt{2}}{5} by 4.

$$n=-\frac{7\sqrt{2}}{10}$$

There are two solutions for n: \frac{\sqrt{2}}{10} and -\frac{7\sqrt{2}}{10}. Substitute \frac{\sqrt{2}}{10} for n in the equation m=n+\frac{3\sqrt{2}}{5} to find the corresponding solution for m that satisfies both equations.

$$m=\frac{\sqrt{2}}{10}+\frac{3\sqrt{2}}{5}$$

Now substitute -\frac{7\sqrt{2}}{10} for n in the equation m=n+\frac{3\sqrt{2}}{5} and solve to find the corresponding solution for m that satisfies both equations.

$$m=-\frac{7\sqrt{2}}{10}+\frac{3\sqrt{2}}{5}$$

Add 1\left(-\frac{7\sqrt{2}}{10}\right) to \frac{3\sqrt{2}}{5}.

$$m=\frac{3\sqrt{2}}{5}-\frac{7\sqrt{2}}{10}$$

The system is now solved.

$$m=\frac{\sqrt{2}}{10}+\frac{3\sqrt{2}}{5},n=\frac{\sqrt{2}}{10}\text{ or }m=\frac{3\sqrt{2}}{5}-\frac{7\sqrt{2}}{10},n=-\frac{7\sqrt{2}}{10}$$