Solve {l}{x^2-y^2/2=1}{3x+2y-1=0}

$$\left\{ \begin{array} { l } { \frac { x ^ { 2 } - y ^ { 2 } } { 2 } = 1 } \\ { 3 x + 2 y - 1 = 0 } \end{array} \right.$$

Solve for x, y (complex solution)

$$x=\frac{3}{5}+\frac{6}{5}i=0.6+1.2i\text{, }y=-\frac{2}{5}-\frac{9}{5}i=-0.4-1.8i$$

$$x=\frac{3}{5}-\frac{6}{5}i=0.6-1.2i\text{, }y=-\frac{2}{5}+\frac{9}{5}i=-0.4+1.8i$$

Steps Using Substitution and Quadratic Formula

Consider the first equation. Multiply both sides by 2.

$$x^{2}-y^{2}=2$$

Consider the second equation. Add 1 to both sides. Anything plus zero gives itself.

$$3x+2y=1$$

Subtract 2y from both sides of the equation.

$$3x=-2y+1$$

Divide both sides by 3.

$$x=-\frac{2}{3}y+\frac{1}{3}$$

Substitute -\frac{2}{3}y+\frac{1}{3} for x in the other equation, -y^{2}+x^{2}=2.

$$-y^{2}+\left(-\frac{2}{3}y+\frac{1}{3}\right)^{2}=2$$

Square -\frac{2}{3}y+\frac{1}{3}.

$$-y^{2}+\frac{4}{9}y^{2}-\frac{4}{9}y+\frac{1}{9}=2$$

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

$$-\frac{5}{9}y^{2}-\frac{4}{9}y+\frac{1}{9}=2$$

Subtract 2 from both sides of the equation.

$$-\frac{5}{9}y^{2}-\frac{4}{9}y-\frac{17}{9}=0$$

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{1}{3}\left(-\frac{2}{3}\right)\times 2 for b, and -\frac{17}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-\frac{4}{9}\right)±\sqrt{\left(-\frac{4}{9}\right)^{2}-4\left(-\frac{5}{9}\right)\left(-\frac{17}{9}\right)}}{2\left(-\frac{5}{9}\right)}$$

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

$$y=\frac{-\left(-\frac{4}{9}\right)±\sqrt{\frac{16}{81}-4\left(-\frac{5}{9}\right)\left(-\frac{17}{9}\right)}}{2\left(-\frac{5}{9}\right)}$$

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

$$y=\frac{-\left(-\frac{4}{9}\right)±\sqrt{\frac{16}{81}+\frac{20}{9}\left(-\frac{17}{9}\right)}}{2\left(-\frac{5}{9}\right)}$$

Multiply \frac{20}{9} times -\frac{17}{9} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{4}{9}\right)±\sqrt{\frac{16-340}{81}}}{2\left(-\frac{5}{9}\right)}$$

Add \frac{16}{81} to -\frac{340}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{4}{9}\right)±\sqrt{-4}}{2\left(-\frac{5}{9}\right)}$$

Take the square root of -4.

$$y=\frac{-\left(-\frac{4}{9}\right)±2i}{2\left(-\frac{5}{9}\right)}$$

The opposite of 1\times \frac{1}{3}\left(-\frac{2}{3}\right)\times 2 is \frac{4}{9}.

$$y=\frac{\frac{4}{9}±2i}{2\left(-\frac{5}{9}\right)}$$

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

$$y=\frac{\frac{4}{9}±2i}{-\frac{10}{9}}$$

Now solve the equation y=\frac{\frac{4}{9}±2i}{-\frac{10}{9}} when ± is plus. Add \frac{4}{9} to 2i.

$$y=\frac{\frac{4}{9}+2i}{-\frac{10}{9}}$$

Divide \frac{4}{9}+2i by -\frac{10}{9} by multiplying \frac{4}{9}+2i by the reciprocal of -\frac{10}{9}.

$$y=-\frac{2}{5}-\frac{9}{5}i$$

Now solve the equation y=\frac{\frac{4}{9}±2i}{-\frac{10}{9}} when ± is minus. Subtract 2i from \frac{4}{9}.

$$y=\frac{\frac{4}{9}-2i}{-\frac{10}{9}}$$

Divide \frac{4}{9}-2i by -\frac{10}{9} by multiplying \frac{4}{9}-2i by the reciprocal of -\frac{10}{9}.

$$y=-\frac{2}{5}+\frac{9}{5}i$$

There are two solutions for y: -\frac{2}{5}-\frac{9}{5}i and -\frac{2}{5}+\frac{9}{5}i. Substitute -\frac{2}{5}-\frac{9}{5}i for y in the equation x=-\frac{2}{3}y+\frac{1}{3} to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{2}{3}\left(-\frac{2}{5}-\frac{9}{5}i\right)+\frac{1}{3}$$

Multiply -\frac{2}{3} times -\frac{2}{5}-\frac{9}{5}i.

$$x=\frac{4}{15}+\frac{6}{5}i+\frac{1}{3}$$

Add -\frac{2}{3}\left(-\frac{2}{5}-\frac{9}{5}i\right) to \frac{1}{3}.

$$x=\frac{3}{5}+\frac{6}{5}i$$

Now substitute -\frac{2}{5}+\frac{9}{5}i for y in the equation x=-\frac{2}{3}y+\frac{1}{3} and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{2}{3}\left(-\frac{2}{5}+\frac{9}{5}i\right)+\frac{1}{3}$$

Multiply -\frac{2}{3} times -\frac{2}{5}+\frac{9}{5}i.

$$x=\frac{4}{15}-\frac{6}{5}i+\frac{1}{3}$$

Add -\frac{2}{3}\left(-\frac{2}{5}+\frac{9}{5}i\right) to \frac{1}{3}.

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

The system is now solved.

$$x=\frac{3}{5}+\frac{6}{5}i,y=-\frac{2}{5}-\frac{9}{5}i\text{ or }x=\frac{3}{5}-\frac{6}{5}i,y=-\frac{2}{5}+\frac{9}{5}i$$