Solve {l}{5x^2+3y^2=192}{5x-3y=-6}

$$\left\{ \begin{array} { l } { 5 x ^ { 2 } + 3 y ^ { 2 } = 192 } \\ { 5 x - 3 y = - 6 } \end{array} \right.$$

Solve for x, y

$$x=3\text{, }y=7$$

$$x=-\frac{9}{2}=-4.5\text{, }y=-\frac{11}{2}=-5.5$$

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.

$$5x-3y=-6,3y^{2}+5x^{2}=192$$

Solve 5x-3y=-6 for x by isolating x on the left hand side of the equal sign.

$$5x-3y=-6$$

Subtract -3y from both sides of the equation.

$$5x=3y-6$$

Divide both sides by 5.

$$x=\frac{3}{5}y-\frac{6}{5}$$

Substitute \frac{3}{5}y-\frac{6}{5} for x in the other equation, 3y^{2}+5x^{2}=192.

$$3y^{2}+5\left(\frac{3}{5}y-\frac{6}{5}\right)^{2}=192$$

Square \frac{3}{5}y-\frac{6}{5}.

$$3y^{2}+5\left(\frac{9}{25}y^{2}-\frac{36}{25}y+\frac{36}{25}\right)=192$$

Multiply 5 times \frac{9}{25}y^{2}-\frac{36}{25}y+\frac{36}{25}.

$$3y^{2}+\frac{9}{5}y^{2}-\frac{36}{5}y+\frac{36}{5}=192$$

Add 3y^{2} to \frac{9}{5}y^{2}.

$$\frac{24}{5}y^{2}-\frac{36}{5}y+\frac{36}{5}=192$$

Subtract 192 from both sides of the equation.

$$\frac{24}{5}y^{2}-\frac{36}{5}y-\frac{924}{5}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3+5\times \left(\frac{3}{5}\right)^{2} for a, 5\left(-\frac{6}{5}\right)\times \frac{3}{5}\times 2 for b, and -\frac{924}{5} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-\frac{36}{5}\right)±\sqrt{\left(-\frac{36}{5}\right)^{2}-4\times \frac{24}{5}\left(-\frac{924}{5}\right)}}{2\times \frac{24}{5}}$$

Square 5\left(-\frac{6}{5}\right)\times \frac{3}{5}\times 2.

$$y=\frac{-\left(-\frac{36}{5}\right)±\sqrt{\frac{1296}{25}-4\times \frac{24}{5}\left(-\frac{924}{5}\right)}}{2\times \frac{24}{5}}$$

Multiply -4 times 3+5\times \left(\frac{3}{5}\right)^{2}.

$$y=\frac{-\left(-\frac{36}{5}\right)±\sqrt{\frac{1296}{25}-\frac{96}{5}\left(-\frac{924}{5}\right)}}{2\times \frac{24}{5}}$$

Multiply -\frac{96}{5} times -\frac{924}{5} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{36}{5}\right)±\sqrt{\frac{1296+88704}{25}}}{2\times \frac{24}{5}}$$

Add \frac{1296}{25} to \frac{88704}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{36}{5}\right)±\sqrt{3600}}{2\times \frac{24}{5}}$$

Take the square root of 3600.

$$y=\frac{-\left(-\frac{36}{5}\right)±60}{2\times \frac{24}{5}}$$

The opposite of 5\left(-\frac{6}{5}\right)\times \frac{3}{5}\times 2 is \frac{36}{5}.

$$y=\frac{\frac{36}{5}±60}{2\times \frac{24}{5}}$$

Multiply 2 times 3+5\times \left(\frac{3}{5}\right)^{2}.

$$y=\frac{\frac{36}{5}±60}{\frac{48}{5}}$$

Now solve the equation y=\frac{\frac{36}{5}±60}{\frac{48}{5}} when ± is plus. Add \frac{36}{5} to 60.

$$y=\frac{\frac{336}{5}}{\frac{48}{5}}$$

Divide \frac{336}{5} by \frac{48}{5} by multiplying \frac{336}{5} by the reciprocal of \frac{48}{5}.

$$y=7$$

Now solve the equation y=\frac{\frac{36}{5}±60}{\frac{48}{5}} when ± is minus. Subtract 60 from \frac{36}{5}.

$$y=-\frac{\frac{264}{5}}{\frac{48}{5}}$$

Divide -\frac{264}{5} by \frac{48}{5} by multiplying -\frac{264}{5} by the reciprocal of \frac{48}{5}.

$$y=-\frac{11}{2}$$

There are two solutions for y: 7 and -\frac{11}{2}. Substitute 7 for y in the equation x=\frac{3}{5}y-\frac{6}{5} to find the corresponding solution for x that satisfies both equations.

$$x=\frac{3}{5}\times 7-\frac{6}{5}$$

Multiply \frac{3}{5} times 7.

$$x=\frac{21-6}{5}$$

Add \frac{3}{5}\times 7 to -\frac{6}{5}.

$$x=3$$

Now substitute -\frac{11}{2} for y in the equation x=\frac{3}{5}y-\frac{6}{5} and solve to find the corresponding solution for x that satisfies both equations.

$$x=\frac{3}{5}\left(-\frac{11}{2}\right)-\frac{6}{5}$$

Multiply \frac{3}{5} times -\frac{11}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{33}{10}-\frac{6}{5}$$

Add -\frac{11}{2}\times \frac{3}{5} to -\frac{6}{5}.

$$x=-\frac{9}{2}$$

The system is now solved.

$$x=3,y=7\text{ or }x=-\frac{9}{2},y=-\frac{11}{2}$$