Solve {l}{x^2/25+y^2/9=1}{4x-5y+40=0}

$$\left. \begin{array} { l } { \frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1 } \\ { 4 x - 5 y + 40 = 0 } \end{array} \right.$$

Solve for x, y (complex solution)

$$x=\frac{-3\sqrt{39}i-32}{5}\approx -6.4-3.746998799i\text{, }y=\frac{-12\sqrt{39}i+72}{25}\approx 2.88-2.997599039i$$

$$x=\frac{-32+3\sqrt{39}i}{5}\approx -6.4+3.746998799i\text{, }y=\frac{72+12\sqrt{39}i}{25}\approx 2.88+2.997599039i$$

Steps Using Substitution and Quadratic Formula

Consider the first equation. Multiply both sides of the equation by 225, the least common multiple of 25,9.

$$9x^{2}+25y^{2}=225$$

Consider the second equation. Subtract 40 from both sides. Anything subtracted from zero gives its negation.

$$4x-5y=-40$$

Subtract -5y from both sides of the equation.

$$4x=5y-40$$

Divide both sides by 4.

$$x=\frac{5}{4}y-10$$

Substitute \frac{5}{4}y-10 for x in the other equation, 25y^{2}+9x^{2}=225.

$$25y^{2}+9\left(\frac{5}{4}y-10\right)^{2}=225$$

Square \frac{5}{4}y-10.

$$25y^{2}+9\left(\frac{25}{16}y^{2}-25y+100\right)=225$$

Multiply 9 times \frac{25}{16}y^{2}-25y+100.

$$25y^{2}+\frac{225}{16}y^{2}-225y+900=225$$

Add 25y^{2} to \frac{225}{16}y^{2}.

$$\frac{625}{16}y^{2}-225y+900=225$$

Subtract 225 from both sides of the equation.

$$\frac{625}{16}y^{2}-225y+675=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 25+9\times \left(\frac{5}{4}\right)^{2} for a, 9\left(-10\right)\times \frac{5}{4}\times 2 for b, and 675 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-225\right)±\sqrt{\left(-225\right)^{2}-4\times \frac{625}{16}\times 675}}{2\times \frac{625}{16}}$$

Square 9\left(-10\right)\times \frac{5}{4}\times 2.

$$y=\frac{-\left(-225\right)±\sqrt{50625-4\times \frac{625}{16}\times 675}}{2\times \frac{625}{16}}$$

Multiply -4 times 25+9\times \left(\frac{5}{4}\right)^{2}.

$$y=\frac{-\left(-225\right)±\sqrt{50625-\frac{625}{4}\times 675}}{2\times \frac{625}{16}}$$

Multiply -\frac{625}{4} times 675.

$$y=\frac{-\left(-225\right)±\sqrt{50625-\frac{421875}{4}}}{2\times \frac{625}{16}}$$

Add 50625 to -\frac{421875}{4}.

$$y=\frac{-\left(-225\right)±\sqrt{-\frac{219375}{4}}}{2\times \frac{625}{16}}$$

Take the square root of -\frac{219375}{4}.

$$y=\frac{-\left(-225\right)±\frac{75\sqrt{39}i}{2}}{2\times \frac{625}{16}}$$

The opposite of 9\left(-10\right)\times \frac{5}{4}\times 2 is 225.

$$y=\frac{225±\frac{75\sqrt{39}i}{2}}{2\times \frac{625}{16}}$$

Multiply 2 times 25+9\times \left(\frac{5}{4}\right)^{2}.

$$y=\frac{225±\frac{75\sqrt{39}i}{2}}{\frac{625}{8}}$$

Now solve the equation y=\frac{225±\frac{75\sqrt{39}i}{2}}{\frac{625}{8}} when ± is plus. Add 225 to \frac{75i\sqrt{39}}{2}.

$$y=\frac{\frac{75\sqrt{39}i}{2}+225}{\frac{625}{8}}$$

Divide 225+\frac{75i\sqrt{39}}{2} by \frac{625}{8} by multiplying 225+\frac{75i\sqrt{39}}{2} by the reciprocal of \frac{625}{8}.

$$y=\frac{72+12\sqrt{39}i}{25}$$

Now solve the equation y=\frac{225±\frac{75\sqrt{39}i}{2}}{\frac{625}{8}} when ± is minus. Subtract \frac{75i\sqrt{39}}{2} from 225.

$$y=\frac{-\frac{75\sqrt{39}i}{2}+225}{\frac{625}{8}}$$

Divide 225-\frac{75i\sqrt{39}}{2} by \frac{625}{8} by multiplying 225-\frac{75i\sqrt{39}}{2} by the reciprocal of \frac{625}{8}.

$$y=\frac{-12\sqrt{39}i+72}{25}$$

There are two solutions for y: \frac{72+12i\sqrt{39}}{25} and \frac{72-12i\sqrt{39}}{25}. Substitute \frac{72+12i\sqrt{39}}{25} for y in the equation x=\frac{5}{4}y-10 to find the corresponding solution for x that satisfies both equations.

$$x=\frac{5}{4}\times \frac{72+12\sqrt{39}i}{25}-10$$

Multiply \frac{5}{4} times \frac{72+12i\sqrt{39}}{25}.

$$x=\frac{5\times \frac{72+12\sqrt{39}i}{25}}{4}-10$$

Now substitute \frac{72-12i\sqrt{39}}{25} for y in the equation x=\frac{5}{4}y-10 and solve to find the corresponding solution for x that satisfies both equations.

$$x=\frac{5}{4}\times \frac{-12\sqrt{39}i+72}{25}-10$$

Multiply \frac{5}{4} times \frac{72-12i\sqrt{39}}{25}.

$$x=\frac{5\times \frac{-12\sqrt{39}i+72}{25}}{4}-10$$

The system is now solved.

$$x=\frac{5\times \frac{72+12\sqrt{39}i}{25}}{4}-10,y=\frac{72+12\sqrt{39}i}{25}\text{ or }x=\frac{5\times \frac{-12\sqrt{39}i+72}{25}}{4}-10,y=\frac{-12\sqrt{39}i+72}{25}$$