Solve {l}{4x+3y=-25/13}{x^2+y^2=1}

$$\left\{ \begin{array} { l } { 4 x + 3 y = - \frac { 25 } { 13 } } \\ { x ^ { 2 } + y ^ { 2 } = 1 } \end{array} \right.$$

Solve for x, y

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

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

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.

$$4x+3y=-\frac{25}{13},y^{2}+x^{2}=1$$

Solve 4x+3y=-\frac{25}{13} for x by isolating x on the left hand side of the equal sign.

$$4x+3y=-\frac{25}{13}$$

Subtract 3y from both sides of the equation.

$$4x=-3y-\frac{25}{13}$$

Divide both sides by 4.

$$x=-\frac{3}{4}y-\frac{25}{52}$$

Substitute -\frac{3}{4}y-\frac{25}{52} for x in the other equation, y^{2}+x^{2}=1.

$$y^{2}+\left(-\frac{3}{4}y-\frac{25}{52}\right)^{2}=1$$

Square -\frac{3}{4}y-\frac{25}{52}.

$$y^{2}+\frac{9}{16}y^{2}+\frac{75}{104}y+\frac{625}{2704}=1$$

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

$$\frac{25}{16}y^{2}+\frac{75}{104}y+\frac{625}{2704}=1$$

Subtract 1 from both sides of the equation.

$$\frac{25}{16}y^{2}+\frac{75}{104}y-\frac{2079}{2704}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\left(-\frac{3}{4}\right)^{2} for a, 1\left(-\frac{25}{52}\right)\left(-\frac{3}{4}\right)\times 2 for b, and -\frac{2079}{2704} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\frac{75}{104}±\sqrt{\left(\frac{75}{104}\right)^{2}-4\times \frac{25}{16}\left(-\frac{2079}{2704}\right)}}{2\times \frac{25}{16}}$$

Square 1\left(-\frac{25}{52}\right)\left(-\frac{3}{4}\right)\times 2.

$$y=\frac{-\frac{75}{104}±\sqrt{\frac{5625}{10816}-4\times \frac{25}{16}\left(-\frac{2079}{2704}\right)}}{2\times \frac{25}{16}}$$

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

$$y=\frac{-\frac{75}{104}±\sqrt{\frac{5625}{10816}-\frac{25}{4}\left(-\frac{2079}{2704}\right)}}{2\times \frac{25}{16}}$$

Multiply -\frac{25}{4} times -\frac{2079}{2704} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\frac{75}{104}±\sqrt{\frac{5625+51975}{10816}}}{2\times \frac{25}{16}}$$

Add \frac{5625}{10816} to \frac{51975}{10816} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\frac{75}{104}±\sqrt{\frac{900}{169}}}{2\times \frac{25}{16}}$$

Take the square root of \frac{900}{169}.

$$y=\frac{-\frac{75}{104}±\frac{30}{13}}{2\times \frac{25}{16}}$$

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

$$y=\frac{-\frac{75}{104}±\frac{30}{13}}{\frac{25}{8}}$$

Now solve the equation y=\frac{-\frac{75}{104}±\frac{30}{13}}{\frac{25}{8}} when ± is plus. Add -\frac{75}{104} to \frac{30}{13} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{\frac{165}{104}}{\frac{25}{8}}$$

Divide \frac{165}{104} by \frac{25}{8} by multiplying \frac{165}{104} by the reciprocal of \frac{25}{8}.

$$y=\frac{33}{65}$$

Now solve the equation y=\frac{-\frac{75}{104}±\frac{30}{13}}{\frac{25}{8}} when ± is minus. Subtract \frac{30}{13} from -\frac{75}{104} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$y=-\frac{\frac{315}{104}}{\frac{25}{8}}$$

Divide -\frac{315}{104} by \frac{25}{8} by multiplying -\frac{315}{104} by the reciprocal of \frac{25}{8}.

$$y=-\frac{63}{65}$$

There are two solutions for y: \frac{33}{65} and -\frac{63}{65}. Substitute \frac{33}{65} for y in the equation x=-\frac{3}{4}y-\frac{25}{52} to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{3}{4}\times \frac{33}{65}-\frac{25}{52}$$

Multiply -\frac{3}{4} times \frac{33}{65} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{99}{260}-\frac{25}{52}$$

Add -\frac{3}{4}\times \frac{33}{65} to -\frac{25}{52}.

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

Now substitute -\frac{63}{65} for y in the equation x=-\frac{3}{4}y-\frac{25}{52} and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{3}{4}\left(-\frac{63}{65}\right)-\frac{25}{52}$$

Multiply -\frac{3}{4} times -\frac{63}{65} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{189}{260}-\frac{25}{52}$$

Add -\frac{63}{65}\left(-\frac{3}{4}\right) to -\frac{25}{52}.

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

The system is now solved.

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