Solve {l}{3x+4y=1}{x^2+y^2=1}

$$\left. \begin{array} { l } { 3 x + 4 y = 1 } \\ { x ^ { 2 } + y ^ { 2 } = 1 } \end{array} \right.$$

Solve for x, y

$$x=\frac{8\sqrt{6}+3}{25}\approx 0.903836718\text{, }y=\frac{4-6\sqrt{6}}{25}\approx -0.427877538$$

$$x=\frac{3-8\sqrt{6}}{25}\approx -0.663836718\text{, }y=\frac{6\sqrt{6}+4}{25}\approx 0.747877538$$

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.

$$3x+4y=1,y^{2}+x^{2}=1$$

Solve 3x+4y=1 for x by isolating x on the left hand side of the equal sign.

$$3x+4y=1$$

Subtract 4y from both sides of the equation.

$$3x=-4y+1$$

Divide both sides by 3.

$$x=-\frac{4}{3}y+\frac{1}{3}$$

Substitute -\frac{4}{3}y+\frac{1}{3} for x in the other equation, y^{2}+x^{2}=1.

$$y^{2}+\left(-\frac{4}{3}y+\frac{1}{3}\right)^{2}=1$$

Square -\frac{4}{3}y+\frac{1}{3}.

$$y^{2}+\frac{16}{9}y^{2}-\frac{8}{9}y+\frac{1}{9}=1$$

Add y^{2} to \frac{16}{9}y^{2}.

$$\frac{25}{9}y^{2}-\frac{8}{9}y+\frac{1}{9}=1$$

Subtract 1 from both sides of the equation.

$$\frac{25}{9}y^{2}-\frac{8}{9}y-\frac{8}{9}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\left(-\frac{4}{3}\right)^{2} for a, 1\times \frac{1}{3}\left(-\frac{4}{3}\right)\times 2 for b, and -\frac{8}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-\frac{8}{9}\right)±\sqrt{\left(-\frac{8}{9}\right)^{2}-4\times \frac{25}{9}\left(-\frac{8}{9}\right)}}{2\times \frac{25}{9}}$$

Square 1\times \frac{1}{3}\left(-\frac{4}{3}\right)\times 2.

$$y=\frac{-\left(-\frac{8}{9}\right)±\sqrt{\frac{64}{81}-4\times \frac{25}{9}\left(-\frac{8}{9}\right)}}{2\times \frac{25}{9}}$$

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

$$y=\frac{-\left(-\frac{8}{9}\right)±\sqrt{\frac{64}{81}-\frac{100}{9}\left(-\frac{8}{9}\right)}}{2\times \frac{25}{9}}$$

Multiply -\frac{100}{9} times -\frac{8}{9} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{8}{9}\right)±\sqrt{\frac{64+800}{81}}}{2\times \frac{25}{9}}$$

Add \frac{64}{81} to \frac{800}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{8}{9}\right)±\sqrt{\frac{32}{3}}}{2\times \frac{25}{9}}$$

Take the square root of \frac{32}{3}.

$$y=\frac{-\left(-\frac{8}{9}\right)±\frac{4\sqrt{6}}{3}}{2\times \frac{25}{9}}$$

The opposite of 1\times \frac{1}{3}\left(-\frac{4}{3}\right)\times 2 is \frac{8}{9}.

$$y=\frac{\frac{8}{9}±\frac{4\sqrt{6}}{3}}{2\times \frac{25}{9}}$$

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

$$y=\frac{\frac{8}{9}±\frac{4\sqrt{6}}{3}}{\frac{50}{9}}$$

Now solve the equation y=\frac{\frac{8}{9}±\frac{4\sqrt{6}}{3}}{\frac{50}{9}} when ± is plus. Add \frac{8}{9} to \frac{4\sqrt{6}}{3}.

$$y=\frac{\frac{4\sqrt{6}}{3}+\frac{8}{9}}{\frac{50}{9}}$$

Divide \frac{8}{9}+\frac{4\sqrt{6}}{3} by \frac{50}{9} by multiplying \frac{8}{9}+\frac{4\sqrt{6}}{3} by the reciprocal of \frac{50}{9}.

$$y=\frac{6\sqrt{6}+4}{25}$$

Now solve the equation y=\frac{\frac{8}{9}±\frac{4\sqrt{6}}{3}}{\frac{50}{9}} when ± is minus. Subtract \frac{4\sqrt{6}}{3} from \frac{8}{9}.

$$y=\frac{-\frac{4\sqrt{6}}{3}+\frac{8}{9}}{\frac{50}{9}}$$

Divide \frac{8}{9}-\frac{4\sqrt{6}}{3} by \frac{50}{9} by multiplying \frac{8}{9}-\frac{4\sqrt{6}}{3} by the reciprocal of \frac{50}{9}.

$$y=\frac{4-6\sqrt{6}}{25}$$

There are two solutions for y: \frac{4+6\sqrt{6}}{25} and \frac{4-6\sqrt{6}}{25}. Substitute \frac{4+6\sqrt{6}}{25} for y in the equation x=-\frac{4}{3}y+\frac{1}{3} to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{4}{3}\times \frac{6\sqrt{6}+4}{25}+\frac{1}{3}$$

Multiply -\frac{4}{3} times \frac{4+6\sqrt{6}}{25}.

$$x=\frac{-4\times \frac{6\sqrt{6}+4}{25}+1}{3}$$

Now substitute \frac{4-6\sqrt{6}}{25} for y in the equation x=-\frac{4}{3}y+\frac{1}{3} and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{4}{3}\times \frac{4-6\sqrt{6}}{25}+\frac{1}{3}$$

Multiply -\frac{4}{3} times \frac{4-6\sqrt{6}}{25}.

$$x=\frac{-4\times \frac{4-6\sqrt{6}}{25}+1}{3}$$

The system is now solved.

$$x=\frac{-4\times \frac{6\sqrt{6}+4}{25}+1}{3},y=\frac{6\sqrt{6}+4}{25}\text{ or }x=\frac{-4\times \frac{4-6\sqrt{6}}{25}+1}{3},y=\frac{4-6\sqrt{6}}{25}$$