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

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

Solve for x, y

$$x=\frac{3\sqrt{6}}{10}+\frac{1}{5}\approx 0.934846923\text{, }y=-\frac{\sqrt{6}}{10}+\frac{3}{5}\approx 0.355051026$$

$$x=-\frac{3\sqrt{6}}{10}+\frac{1}{5}\approx -0.534846923\text{, }y=\frac{\sqrt{6}}{10}+\frac{3}{5}\approx 0.844948974$$

Steps Using Substitution and Quadratic Formula

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.

$$3y+x=2,x^{2}+y^{2}=1$$

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

$$3y+x=2$$

Subtract x from both sides of the equation.

$$3y=-x+2$$

Divide both sides by 3.

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

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

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

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

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

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

$$\frac{10}{9}x^{2}-\frac{4}{9}x+\frac{4}{9}=1$$

Subtract 1 from both sides of the equation.

$$\frac{10}{9}x^{2}-\frac{4}{9}x-\frac{5}{9}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\left(-\frac{1}{3}\right)^{2} for a, 1\times \frac{2}{3}\left(-\frac{1}{3}\right)\times 2 for b, and -\frac{5}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{4}{9}\right)±\sqrt{\left(-\frac{4}{9}\right)^{2}-4\times \frac{10}{9}\left(-\frac{5}{9}\right)}}{2\times \frac{10}{9}}$$

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

$$x=\frac{-\left(-\frac{4}{9}\right)±\sqrt{\frac{16}{81}-4\times \frac{10}{9}\left(-\frac{5}{9}\right)}}{2\times \frac{10}{9}}$$

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

$$x=\frac{-\left(-\frac{4}{9}\right)±\sqrt{\frac{16}{81}-\frac{40}{9}\left(-\frac{5}{9}\right)}}{2\times \frac{10}{9}}$$

Multiply -\frac{40}{9} times -\frac{5}{9} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{4}{9}\right)±\sqrt{\frac{16+200}{81}}}{2\times \frac{10}{9}}$$

Add \frac{16}{81} to \frac{200}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{4}{9}\right)±\sqrt{\frac{8}{3}}}{2\times \frac{10}{9}}$$

Take the square root of \frac{8}{3}.

$$x=\frac{-\left(-\frac{4}{9}\right)±\frac{2\sqrt{6}}{3}}{2\times \frac{10}{9}}$$

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

$$x=\frac{\frac{4}{9}±\frac{2\sqrt{6}}{3}}{2\times \frac{10}{9}}$$

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

$$x=\frac{\frac{4}{9}±\frac{2\sqrt{6}}{3}}{\frac{20}{9}}$$

Now solve the equation x=\frac{\frac{4}{9}±\frac{2\sqrt{6}}{3}}{\frac{20}{9}} when ± is plus. Add \frac{4}{9} to \frac{2\sqrt{6}}{3}.

$$x=\frac{\frac{2\sqrt{6}}{3}+\frac{4}{9}}{\frac{20}{9}}$$

Divide \frac{4}{9}+\frac{2\sqrt{6}}{3} by \frac{20}{9} by multiplying \frac{4}{9}+\frac{2\sqrt{6}}{3} by the reciprocal of \frac{20}{9}.

$$x=\frac{3\sqrt{6}}{10}+\frac{1}{5}$$

Now solve the equation x=\frac{\frac{4}{9}±\frac{2\sqrt{6}}{3}}{\frac{20}{9}} when ± is minus. Subtract \frac{2\sqrt{6}}{3} from \frac{4}{9}.

$$x=\frac{-\frac{2\sqrt{6}}{3}+\frac{4}{9}}{\frac{20}{9}}$$

Divide \frac{4}{9}-\frac{2\sqrt{6}}{3} by \frac{20}{9} by multiplying \frac{4}{9}-\frac{2\sqrt{6}}{3} by the reciprocal of \frac{20}{9}.

$$x=-\frac{3\sqrt{6}}{10}+\frac{1}{5}$$

There are two solutions for x: \frac{1}{5}+\frac{3\sqrt{6}}{10} and \frac{1}{5}-\frac{3\sqrt{6}}{10}. Substitute \frac{1}{5}+\frac{3\sqrt{6}}{10} for x in the equation y=-\frac{1}{3}x+\frac{2}{3} to find the corresponding solution for y that satisfies both equations.

$$y=-\frac{1}{3}\left(\frac{3\sqrt{6}}{10}+\frac{1}{5}\right)+\frac{2}{3}$$

Multiply -\frac{1}{3} times \frac{1}{5}+\frac{3\sqrt{6}}{10}.

$$y=\frac{-\left(\frac{3\sqrt{6}}{10}+\frac{1}{5}\right)+2}{3}$$

Now substitute \frac{1}{5}-\frac{3\sqrt{6}}{10} for x in the equation y=-\frac{1}{3}x+\frac{2}{3} and solve to find the corresponding solution for y that satisfies both equations.

$$y=-\frac{1}{3}\left(-\frac{3\sqrt{6}}{10}+\frac{1}{5}\right)+\frac{2}{3}$$

Multiply -\frac{1}{3} times \frac{1}{5}-\frac{3\sqrt{6}}{10}.

$$y=\frac{-\left(-\frac{3\sqrt{6}}{10}+\frac{1}{5}\right)+2}{3}$$

The system is now solved.

$$y=\frac{-\left(\frac{3\sqrt{6}}{10}+\frac{1}{5}\right)+2}{3},x=\frac{3\sqrt{6}}{10}+\frac{1}{5}\text{ or }y=\frac{-\left(-\frac{3\sqrt{6}}{10}+\frac{1}{5}\right)+2}{3},x=-\frac{3\sqrt{6}}{10}+\frac{1}{5}$$