Solve {l}{x^2/16-y^2/25=1}{125x-48y=481}

$$\left. \begin{array}{l}{ \frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 25 } = 1 }\\{ 125 x - 48 y = 481 }\end{array} \right.$$

Solve for x, y

$$x=-\frac{108\sqrt{481}}{2405}+5\approx 4.015124774\text{, }y=-\frac{225\sqrt{481}}{1924}+3\approx 0.435220767$$

$$x=\frac{108\sqrt{481}}{2405}+5\approx 5.984875226\text{, }y=\frac{225\sqrt{481}}{1924}+3\approx 5.564779233$$

Steps Using Substitution and Quadratic Formula

Consider the first equation. Multiply both sides of the equation by 400, the least common multiple of 16,25.

$$25x^{2}-16y^{2}=400$$

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.

$$125x-48y=481,-16y^{2}+25x^{2}=400$$

Solve 125x-48y=481 for x by isolating x on the left hand side of the equal sign.

$$125x-48y=481$$

Subtract -48y from both sides of the equation.

$$125x=48y+481$$

Divide both sides by 125.

$$x=\frac{48}{125}y+\frac{481}{125}$$

Substitute \frac{48}{125}y+\frac{481}{125} for x in the other equation, -16y^{2}+25x^{2}=400.

$$-16y^{2}+25\left(\frac{48}{125}y+\frac{481}{125}\right)^{2}=400$$

Square \frac{48}{125}y+\frac{481}{125}.

$$-16y^{2}+25\left(\frac{2304}{15625}y^{2}+\frac{46176}{15625}y+\frac{231361}{15625}\right)=400$$

Multiply 25 times \frac{2304}{15625}y^{2}+\frac{46176}{15625}y+\frac{231361}{15625}.

$$-16y^{2}+\frac{2304}{625}y^{2}+\frac{46176}{625}y+\frac{231361}{625}=400$$

Add -16y^{2} to \frac{2304}{625}y^{2}.

$$-\frac{7696}{625}y^{2}+\frac{46176}{625}y+\frac{231361}{625}=400$$

Subtract 400 from both sides of the equation.

$$-\frac{7696}{625}y^{2}+\frac{46176}{625}y-\frac{18639}{625}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16+25\times \left(\frac{48}{125}\right)^{2} for a, 25\times \frac{481}{125}\times \frac{48}{125}\times 2 for b, and -\frac{18639}{625} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\frac{46176}{625}±\sqrt{\left(\frac{46176}{625}\right)^{2}-4\left(-\frac{7696}{625}\right)\left(-\frac{18639}{625}\right)}}{2\left(-\frac{7696}{625}\right)}$$

Square 25\times \frac{481}{125}\times \frac{48}{125}\times 2.

$$y=\frac{-\frac{46176}{625}±\sqrt{\frac{2132222976}{390625}-4\left(-\frac{7696}{625}\right)\left(-\frac{18639}{625}\right)}}{2\left(-\frac{7696}{625}\right)}$$

Multiply -4 times -16+25\times \left(\frac{48}{125}\right)^{2}.

$$y=\frac{-\frac{46176}{625}±\sqrt{\frac{2132222976}{390625}+\frac{30784}{625}\left(-\frac{18639}{625}\right)}}{2\left(-\frac{7696}{625}\right)}$$

Multiply \frac{30784}{625} times -\frac{18639}{625} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\frac{46176}{625}±\sqrt{\frac{2132222976-573782976}{390625}}}{2\left(-\frac{7696}{625}\right)}$$

Add \frac{2132222976}{390625} to -\frac{573782976}{390625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\frac{46176}{625}±\sqrt{\frac{2493504}{625}}}{2\left(-\frac{7696}{625}\right)}$$

Take the square root of \frac{2493504}{625}.

$$y=\frac{-\frac{46176}{625}±\frac{72\sqrt{481}}{25}}{2\left(-\frac{7696}{625}\right)}$$

Multiply 2 times -16+25\times \left(\frac{48}{125}\right)^{2}.

$$y=\frac{-\frac{46176}{625}±\frac{72\sqrt{481}}{25}}{-\frac{15392}{625}}$$

Now solve the equation y=\frac{-\frac{46176}{625}±\frac{72\sqrt{481}}{25}}{-\frac{15392}{625}} when ± is plus. Add -\frac{46176}{625} to \frac{72\sqrt{481}}{25}.

$$y=\frac{\frac{72\sqrt{481}}{25}-\frac{46176}{625}}{-\frac{15392}{625}}$$

Divide -\frac{46176}{625}+\frac{72\sqrt{481}}{25} by -\frac{15392}{625} by multiplying -\frac{46176}{625}+\frac{72\sqrt{481}}{25} by the reciprocal of -\frac{15392}{625}.

$$y=-\frac{225\sqrt{481}}{1924}+3$$

Now solve the equation y=\frac{-\frac{46176}{625}±\frac{72\sqrt{481}}{25}}{-\frac{15392}{625}} when ± is minus. Subtract \frac{72\sqrt{481}}{25} from -\frac{46176}{625}.

$$y=\frac{-\frac{72\sqrt{481}}{25}-\frac{46176}{625}}{-\frac{15392}{625}}$$

Divide -\frac{46176}{625}-\frac{72\sqrt{481}}{25} by -\frac{15392}{625} by multiplying -\frac{46176}{625}-\frac{72\sqrt{481}}{25} by the reciprocal of -\frac{15392}{625}.

$$y=\frac{225\sqrt{481}}{1924}+3$$

There are two solutions for y: 3-\frac{225\sqrt{481}}{1924} and 3+\frac{225\sqrt{481}}{1924}. Substitute 3-\frac{225\sqrt{481}}{1924} for y in the equation x=\frac{48}{125}y+\frac{481}{125} to find the corresponding solution for x that satisfies both equations.

$$x=\frac{48}{125}\left(-\frac{225\sqrt{481}}{1924}+3\right)+\frac{481}{125}$$

Multiply \frac{48}{125} times 3-\frac{225\sqrt{481}}{1924}.

$$x=\frac{48\left(-\frac{225\sqrt{481}}{1924}+3\right)+481}{125}$$

Now substitute 3+\frac{225\sqrt{481}}{1924} for y in the equation x=\frac{48}{125}y+\frac{481}{125} and solve to find the corresponding solution for x that satisfies both equations.

$$x=\frac{48}{125}\left(\frac{225\sqrt{481}}{1924}+3\right)+\frac{481}{125}$$

Multiply \frac{48}{125} times 3+\frac{225\sqrt{481}}{1924}.

$$x=\frac{48\left(\frac{225\sqrt{481}}{1924}+3\right)+481}{125}$$

The system is now solved.

$$x=\frac{48\left(-\frac{225\sqrt{481}}{1924}+3\right)+481}{125},y=-\frac{225\sqrt{481}}{1924}+3\text{ or }x=\frac{48\left(\frac{225\sqrt{481}}{1924}+3\right)+481}{125},y=\frac{225\sqrt{481}}{1924}+3$$