Solve {l}{x+y=5}{x^2/9+y^2=1}

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

Solve for x, y (complex solution)

$$x=\frac{3\sqrt{15}i}{10}+\frac{9}{2}\approx 4.5+1.161895004i\text{, }y=-\frac{3\sqrt{15}i}{10}+\frac{1}{2}\approx 0.5-1.161895004i$$

$$x=-\frac{3\sqrt{15}i}{10}+\frac{9}{2}\approx 4.5-1.161895004i\text{, }y=\frac{3\sqrt{15}i}{10}+\frac{1}{2}\approx 0.5+1.161895004i$$

Steps Using Substitution and Quadratic Formula

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

$$x+y=5$$

Subtract y from both sides of the equation.

$$x=-y+5$$

Substitute -y+5 for x in the other equation, y^{2}+\frac{1}{9}x^{2}=1.

$$y^{2}+\frac{1}{9}\left(-y+5\right)^{2}=1$$

Square -y+5.

$$y^{2}+\frac{1}{9}\left(y^{2}-10y+25\right)=1$$

Multiply \frac{1}{9} times y^{2}-10y+25.

$$y^{2}+\frac{1}{9}y^{2}-\frac{10}{9}y+\frac{25}{9}=1$$

Add y^{2} to \frac{1}{9}y^{2}.

$$\frac{10}{9}y^{2}-\frac{10}{9}y+\frac{25}{9}=1$$

Subtract 1 from both sides of the equation.

$$\frac{10}{9}y^{2}-\frac{10}{9}y+\frac{16}{9}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+\frac{1}{9}\left(-1\right)^{2} for a, \frac{1}{9}\times 5\left(-1\right)\times 2 for b, and \frac{16}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-\frac{10}{9}\right)±\sqrt{\left(-\frac{10}{9}\right)^{2}-4\times \frac{10}{9}\times \frac{16}{9}}}{2\times \frac{10}{9}}$$

Square \frac{1}{9}\times 5\left(-1\right)\times 2.

$$y=\frac{-\left(-\frac{10}{9}\right)±\sqrt{\frac{100}{81}-4\times \frac{10}{9}\times \frac{16}{9}}}{2\times \frac{10}{9}}$$

Multiply -4 times 1+\frac{1}{9}\left(-1\right)^{2}.

$$y=\frac{-\left(-\frac{10}{9}\right)±\sqrt{\frac{100}{81}-\frac{40}{9}\times \frac{16}{9}}}{2\times \frac{10}{9}}$$

Multiply -\frac{40}{9} times \frac{16}{9} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{10}{9}\right)±\sqrt{\frac{100-640}{81}}}{2\times \frac{10}{9}}$$

Add \frac{100}{81} to -\frac{640}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{10}{9}\right)±\sqrt{-\frac{20}{3}}}{2\times \frac{10}{9}}$$

Take the square root of -\frac{20}{3}.

$$y=\frac{-\left(-\frac{10}{9}\right)±\frac{2\sqrt{15}i}{3}}{2\times \frac{10}{9}}$$

The opposite of \frac{1}{9}\times 5\left(-1\right)\times 2 is \frac{10}{9}.

$$y=\frac{\frac{10}{9}±\frac{2\sqrt{15}i}{3}}{2\times \frac{10}{9}}$$

Multiply 2 times 1+\frac{1}{9}\left(-1\right)^{2}.

$$y=\frac{\frac{10}{9}±\frac{2\sqrt{15}i}{3}}{\frac{20}{9}}$$

Now solve the equation y=\frac{\frac{10}{9}±\frac{2\sqrt{15}i}{3}}{\frac{20}{9}} when ± is plus. Add \frac{10}{9} to \frac{2i\sqrt{15}}{3}.

$$y=\frac{\frac{2\sqrt{15}i}{3}+\frac{10}{9}}{\frac{20}{9}}$$

Divide \frac{10}{9}+\frac{2i\sqrt{15}}{3} by \frac{20}{9} by multiplying \frac{10}{9}+\frac{2i\sqrt{15}}{3} by the reciprocal of \frac{20}{9}.

$$y=\frac{3\sqrt{15}i}{10}+\frac{1}{2}$$

Now solve the equation y=\frac{\frac{10}{9}±\frac{2\sqrt{15}i}{3}}{\frac{20}{9}} when ± is minus. Subtract \frac{2i\sqrt{15}}{3} from \frac{10}{9}.

$$y=\frac{-\frac{2\sqrt{15}i}{3}+\frac{10}{9}}{\frac{20}{9}}$$

Divide \frac{10}{9}-\frac{2i\sqrt{15}}{3} by \frac{20}{9} by multiplying \frac{10}{9}-\frac{2i\sqrt{15}}{3} by the reciprocal of \frac{20}{9}.

$$y=-\frac{3\sqrt{15}i}{10}+\frac{1}{2}$$

There are two solutions for y: \frac{1}{2}+\frac{3i\sqrt{15}}{10} and \frac{1}{2}-\frac{3i\sqrt{15}}{10}. Substitute \frac{1}{2}+\frac{3i\sqrt{15}}{10} for y in the equation x=-y+5 to find the corresponding solution for x that satisfies both equations.

$$x=-\left(\frac{3\sqrt{15}i}{10}+\frac{1}{2}\right)+5$$

Now substitute \frac{1}{2}-\frac{3i\sqrt{15}}{10} for y in the equation x=-y+5 and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\left(-\frac{3\sqrt{15}i}{10}+\frac{1}{2}\right)+5$$

The system is now solved.

$$x=-\left(\frac{3\sqrt{15}i}{10}+\frac{1}{2}\right)+5,y=\frac{3\sqrt{15}i}{10}+\frac{1}{2}\text{ or }x=-\left(-\frac{3\sqrt{15}i}{10}+\frac{1}{2}\right)+5,y=-\frac{3\sqrt{15}i}{10}+\frac{1}{2}$$