Solve {l}{x^2+y^2=3}{3x+2y=11}

$$\left. \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = 3 } \\ { 3 x + 2 y = 11 } \end{array} \right.$$

Solve for x, y (complex solution)

$$x=\frac{33+2\sqrt{82}i}{13}\approx 2.538461538+1.393136175i\text{, }y=\frac{-3\sqrt{82}i+22}{13}\approx 1.692307692-2.089704263i$$

$$x=\frac{-2\sqrt{82}i+33}{13}\approx 2.538461538-1.393136175i\text{, }y=\frac{22+3\sqrt{82}i}{13}\approx 1.692307692+2.089704263i$$

Steps Using Substitution and Quadratic Formula

Solve 3x+2y=11 for x by isolating x on the left hand side of the equal sign.

$$3x+2y=11$$

Subtract 2y from both sides of the equation.

$$3x=-2y+11$$

Divide both sides by 3.

$$x=-\frac{2}{3}y+\frac{11}{3}$$

Substitute -\frac{2}{3}y+\frac{11}{3} for x in the other equation, y^{2}+x^{2}=3.

$$y^{2}+\left(-\frac{2}{3}y+\frac{11}{3}\right)^{2}=3$$

Square -\frac{2}{3}y+\frac{11}{3}.

$$y^{2}+\frac{4}{9}y^{2}-\frac{44}{9}y+\frac{121}{9}=3$$

Add y^{2} to \frac{4}{9}y^{2}.

$$\frac{13}{9}y^{2}-\frac{44}{9}y+\frac{121}{9}=3$$

Subtract 3 from both sides of the equation.

$$\frac{13}{9}y^{2}-\frac{44}{9}y+\frac{94}{9}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\left(-\frac{2}{3}\right)^{2} for a, 1\times \frac{11}{3}\left(-\frac{2}{3}\right)\times 2 for b, and \frac{94}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-\frac{44}{9}\right)±\sqrt{\left(-\frac{44}{9}\right)^{2}-4\times \frac{13}{9}\times \frac{94}{9}}}{2\times \frac{13}{9}}$$

Square 1\times \frac{11}{3}\left(-\frac{2}{3}\right)\times 2.

$$y=\frac{-\left(-\frac{44}{9}\right)±\sqrt{\frac{1936}{81}-4\times \frac{13}{9}\times \frac{94}{9}}}{2\times \frac{13}{9}}$$

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

$$y=\frac{-\left(-\frac{44}{9}\right)±\sqrt{\frac{1936}{81}-\frac{52}{9}\times \frac{94}{9}}}{2\times \frac{13}{9}}$$

Multiply -\frac{52}{9} times \frac{94}{9} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{44}{9}\right)±\sqrt{\frac{1936-4888}{81}}}{2\times \frac{13}{9}}$$

Add \frac{1936}{81} to -\frac{4888}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{44}{9}\right)±\sqrt{-\frac{328}{9}}}{2\times \frac{13}{9}}$$

Take the square root of -\frac{328}{9}.

$$y=\frac{-\left(-\frac{44}{9}\right)±\frac{2\sqrt{82}i}{3}}{2\times \frac{13}{9}}$$

The opposite of 1\times \frac{11}{3}\left(-\frac{2}{3}\right)\times 2 is \frac{44}{9}.

$$y=\frac{\frac{44}{9}±\frac{2\sqrt{82}i}{3}}{2\times \frac{13}{9}}$$

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

$$y=\frac{\frac{44}{9}±\frac{2\sqrt{82}i}{3}}{\frac{26}{9}}$$

Now solve the equation y=\frac{\frac{44}{9}±\frac{2\sqrt{82}i}{3}}{\frac{26}{9}} when ± is plus. Add \frac{44}{9} to \frac{2i\sqrt{82}}{3}.

$$y=\frac{\frac{2\sqrt{82}i}{3}+\frac{44}{9}}{\frac{26}{9}}$$

Divide \frac{44}{9}+\frac{2i\sqrt{82}}{3} by \frac{26}{9} by multiplying \frac{44}{9}+\frac{2i\sqrt{82}}{3} by the reciprocal of \frac{26}{9}.

$$y=\frac{22+3\sqrt{82}i}{13}$$

Now solve the equation y=\frac{\frac{44}{9}±\frac{2\sqrt{82}i}{3}}{\frac{26}{9}} when ± is minus. Subtract \frac{2i\sqrt{82}}{3} from \frac{44}{9}.

$$y=\frac{-\frac{2\sqrt{82}i}{3}+\frac{44}{9}}{\frac{26}{9}}$$

Divide \frac{44}{9}-\frac{2i\sqrt{82}}{3} by \frac{26}{9} by multiplying \frac{44}{9}-\frac{2i\sqrt{82}}{3} by the reciprocal of \frac{26}{9}.

$$y=\frac{-3\sqrt{82}i+22}{13}$$

There are two solutions for y: \frac{22+3i\sqrt{82}}{13} and \frac{22-3i\sqrt{82}}{13}. Substitute \frac{22+3i\sqrt{82}}{13} for y in the equation x=-\frac{2}{3}y+\frac{11}{3} to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{2}{3}\times \frac{22+3\sqrt{82}i}{13}+\frac{11}{3}$$

Multiply -\frac{2}{3} times \frac{22+3i\sqrt{82}}{13}.

$$x=\frac{-2\times \frac{22+3\sqrt{82}i}{13}+11}{3}$$

Now substitute \frac{22-3i\sqrt{82}}{13} for y in the equation x=-\frac{2}{3}y+\frac{11}{3} and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{2}{3}\times \frac{-3\sqrt{82}i+22}{13}+\frac{11}{3}$$

Multiply -\frac{2}{3} times \frac{22-3i\sqrt{82}}{13}.

$$x=\frac{-2\times \frac{-3\sqrt{82}i+22}{13}+11}{3}$$

The system is now solved.

$$x=\frac{-2\times \frac{22+3\sqrt{82}i}{13}+11}{3},y=\frac{22+3\sqrt{82}i}{13}\text{ or }x=\frac{-2\times \frac{-3\sqrt{82}i+22}{13}+11}{3},y=\frac{-3\sqrt{82}i+22}{13}$$