Solve {l}{3/5x-4/5y=5/13}{x^2+y^2=1}

$$\left\{ \begin{array} { l } { \frac { 3 } { 5 } x - \frac { 4 } { 5 } y = \frac { 5 } { 13 } } \\ { x ^ { 2 } + y ^ { 2 } = 1 } \end{array} \right.$$

Solve for x, y

$$x=\frac{63}{65}\approx 0.969230769\text{, }y=\frac{16}{65}\approx 0.246153846$$

$$x=-\frac{33}{65}\approx -0.507692308\text{, }y=-\frac{56}{65}\approx -0.861538462$$

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.

$$\frac{3}{5}x-\frac{4}{5}y=\frac{5}{13},y^{2}+x^{2}=1$$

Solve \frac{3}{5}x-\frac{4}{5}y=\frac{5}{13} for x by isolating x on the left hand side of the equal sign.

$$\frac{3}{5}x-\frac{4}{5}y=\frac{5}{13}$$

Subtract -\frac{4}{5}y from both sides of the equation.

$$\frac{3}{5}x=\frac{4}{5}y+\frac{5}{13}$$

Divide both sides of the equation by \frac{3}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=\frac{4}{3}y+\frac{25}{39}$$

Substitute \frac{4}{3}y+\frac{25}{39} for x in the other equation, y^{2}+x^{2}=1.

$$y^{2}+\left(\frac{4}{3}y+\frac{25}{39}\right)^{2}=1$$

Square \frac{4}{3}y+\frac{25}{39}.

$$y^{2}+\frac{16}{9}y^{2}+\frac{200}{117}y+\frac{625}{1521}=1$$

Add y^{2} to \frac{16}{9}y^{2}.

$$\frac{25}{9}y^{2}+\frac{200}{117}y+\frac{625}{1521}=1$$

Subtract 1 from both sides of the equation.

$$\frac{25}{9}y^{2}+\frac{200}{117}y-\frac{896}{1521}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\times \left(\frac{4}{3}\right)^{2} for a, 1\times \frac{25}{39}\times \frac{4}{3}\times 2 for b, and -\frac{896}{1521} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\frac{200}{117}±\sqrt{\left(\frac{200}{117}\right)^{2}-4\times \frac{25}{9}\left(-\frac{896}{1521}\right)}}{2\times \frac{25}{9}}$$

Square 1\times \frac{25}{39}\times \frac{4}{3}\times 2.

$$y=\frac{-\frac{200}{117}±\sqrt{\frac{40000}{13689}-4\times \frac{25}{9}\left(-\frac{896}{1521}\right)}}{2\times \frac{25}{9}}$$

Multiply -4 times 1+1\times \left(\frac{4}{3}\right)^{2}.

$$y=\frac{-\frac{200}{117}±\sqrt{\frac{40000}{13689}-\frac{100}{9}\left(-\frac{896}{1521}\right)}}{2\times \frac{25}{9}}$$

Multiply -\frac{100}{9} times -\frac{896}{1521} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\frac{200}{117}±\sqrt{\frac{40000+89600}{13689}}}{2\times \frac{25}{9}}$$

Add \frac{40000}{13689} to \frac{89600}{13689} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\frac{200}{117}±\sqrt{\frac{1600}{169}}}{2\times \frac{25}{9}}$$

Take the square root of \frac{1600}{169}.

$$y=\frac{-\frac{200}{117}±\frac{40}{13}}{2\times \frac{25}{9}}$$

Multiply 2 times 1+1\times \left(\frac{4}{3}\right)^{2}.

$$y=\frac{-\frac{200}{117}±\frac{40}{13}}{\frac{50}{9}}$$

Now solve the equation y=\frac{-\frac{200}{117}±\frac{40}{13}}{\frac{50}{9}} when ± is plus. Add -\frac{200}{117} to \frac{40}{13} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{\frac{160}{117}}{\frac{50}{9}}$$

Divide \frac{160}{117} by \frac{50}{9} by multiplying \frac{160}{117} by the reciprocal of \frac{50}{9}.

$$y=\frac{16}{65}$$

Now solve the equation y=\frac{-\frac{200}{117}±\frac{40}{13}}{\frac{50}{9}} when ± is minus. Subtract \frac{40}{13} from -\frac{200}{117} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$y=-\frac{\frac{560}{117}}{\frac{50}{9}}$$

Divide -\frac{560}{117} by \frac{50}{9} by multiplying -\frac{560}{117} by the reciprocal of \frac{50}{9}.

$$y=-\frac{56}{65}$$

There are two solutions for y: \frac{16}{65} and -\frac{56}{65}. Substitute \frac{16}{65} for y in the equation x=\frac{4}{3}y+\frac{25}{39} to find the corresponding solution for x that satisfies both equations.

$$x=\frac{4}{3}\times \frac{16}{65}+\frac{25}{39}$$

Multiply \frac{4}{3} times \frac{16}{65} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{64}{195}+\frac{25}{39}$$

Add \frac{16}{65}\times \frac{4}{3} to \frac{25}{39}.

$$x=\frac{63}{65}$$

Now substitute -\frac{56}{65} for y in the equation x=\frac{4}{3}y+\frac{25}{39} and solve to find the corresponding solution for x that satisfies both equations.

$$x=\frac{4}{3}\left(-\frac{56}{65}\right)+\frac{25}{39}$$

Multiply \frac{4}{3} times -\frac{56}{65} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{224}{195}+\frac{25}{39}$$

Add -\frac{56}{65}\times \frac{4}{3} to \frac{25}{39}.

$$x=-\frac{33}{65}$$

The system is now solved.

$$x=\frac{63}{65},y=\frac{16}{65}\text{ or }x=-\frac{33}{65},y=-\frac{56}{65}$$