Solve {l}{3/5x+4/5y=frac{2sqrt{5}}{5}}{x^2+y^2=1}

$$\left\{ \begin{array} { l } { \frac { 3 } { 5 } x + \frac { 4 } { 5 } y = \frac { 2 \sqrt { 5 } } { 5 } } \\ { x ^ { 2 } + y ^ { 2 } = 1 } \end{array} \right.$$

Solve for x, y

$$x=\frac{2\sqrt{5}}{5}\approx 0.894427191\text{, }y=\frac{\sqrt{5}}{5}\approx 0.447213595$$

$$x=\frac{2\sqrt{5}}{25}\approx 0.178885438\text{, }y=\frac{11\sqrt{5}}{25}\approx 0.98386991$$

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.

$$\frac{3}{5}x+\frac{4}{5}y=\frac{2\sqrt{5}}{5},y^{2}+x^{2}=1$$

Solve \frac{3}{5}x+\frac{4}{5}y=\frac{2\sqrt{5}}{5} for x by isolating x on the left hand side of the equal sign.

$$\frac{3}{5}x+\frac{4}{5}y=\frac{2\sqrt{5}}{5}$$

Subtract \frac{4}{5}y from both sides of the equation.

$$\frac{3}{5}x=-\frac{4}{5}y+\frac{2\sqrt{5}}{5}$$

Divide both sides of the equation by \frac{3}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=-\frac{4}{3}y+\frac{2\sqrt{5}}{3}$$

Substitute -\frac{4}{3}y+\frac{2\sqrt{5}}{3} for x in the other equation, y^{2}+x^{2}=1.

$$y^{2}+\left(-\frac{4}{3}y+\frac{2\sqrt{5}}{3}\right)^{2}=1$$

Square -\frac{4}{3}y+\frac{2\sqrt{5}}{3}.

$$y^{2}+\frac{16}{9}y^{2}+\left(-\frac{8}{3}\times \frac{2\sqrt{5}}{3}\right)y+\left(\frac{2\sqrt{5}}{3}\right)^{2}=1$$

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

$$\frac{25}{9}y^{2}+\left(-\frac{8}{3}\times \frac{2\sqrt{5}}{3}\right)y+\left(\frac{2\sqrt{5}}{3}\right)^{2}=1$$

Subtract 1 from both sides of the equation.

$$\frac{25}{9}y^{2}+\left(-\frac{8}{3}\times \frac{2\sqrt{5}}{3}\right)y+\left(\frac{2\sqrt{5}}{3}\right)^{2}-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\left(-\frac{4}{3}\right)^{2} for a, 1\left(-\frac{4}{3}\right)\times 2\times \frac{2\sqrt{5}}{3} for b, and \frac{11}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-\frac{8}{3}\times \frac{2\sqrt{5}}{3}\right)±\sqrt{\left(-\frac{8}{3}\times \frac{2\sqrt{5}}{3}\right)^{2}-4\times \frac{25}{9}\times \frac{11}{9}}}{2\times \frac{25}{9}}$$

Square 1\left(-\frac{4}{3}\right)\times 2\times \frac{2\sqrt{5}}{3}.

$$y=\frac{-\left(-\frac{8}{3}\times \frac{2\sqrt{5}}{3}\right)±\sqrt{\frac{1280}{81}-4\times \frac{25}{9}\times \frac{11}{9}}}{2\times \frac{25}{9}}$$

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

$$y=\frac{-\left(-\frac{8}{3}\times \frac{2\sqrt{5}}{3}\right)±\sqrt{\frac{1280}{81}-\frac{100}{9}\times \frac{11}{9}}}{2\times \frac{25}{9}}$$

Multiply -\frac{100}{9} times \frac{11}{9} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{8}{3}\times \frac{2\sqrt{5}}{3}\right)±\sqrt{\frac{1280-1100}{81}}}{2\times \frac{25}{9}}$$

Add \frac{1280}{81} to -\frac{1100}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{8}{3}\times \frac{2\sqrt{5}}{3}\right)±\sqrt{\frac{20}{9}}}{2\times \frac{25}{9}}$$

Take the square root of \frac{20}{9}.

$$y=\frac{-\left(-\frac{8}{3}\times \frac{2\sqrt{5}}{3}\right)±\frac{2\sqrt{5}}{3}}{2\times \frac{25}{9}}$$

The opposite of 1\left(-\frac{4}{3}\right)\times 2\times \frac{2\sqrt{5}}{3} is \frac{16\sqrt{5}}{9}.

$$y=\frac{\frac{16\sqrt{5}}{9}±\frac{2\sqrt{5}}{3}}{2\times \frac{25}{9}}$$

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

$$y=\frac{\frac{16\sqrt{5}}{9}±\frac{2\sqrt{5}}{3}}{\frac{50}{9}}$$

Now solve the equation y=\frac{\frac{16\sqrt{5}}{9}±\frac{2\sqrt{5}}{3}}{\frac{50}{9}} when ± is plus. Add \frac{16\sqrt{5}}{9} to \frac{2\sqrt{5}}{3}.

$$y=\frac{22\sqrt{5}}{\frac{50}{9}\times 9}$$

Divide \frac{22\sqrt{5}}{9} by \frac{50}{9} by multiplying \frac{22\sqrt{5}}{9} by the reciprocal of \frac{50}{9}.

$$y=\frac{11\sqrt{5}}{25}$$

Now solve the equation y=\frac{\frac{16\sqrt{5}}{9}±\frac{2\sqrt{5}}{3}}{\frac{50}{9}} when ± is minus. Subtract \frac{2\sqrt{5}}{3} from \frac{16\sqrt{5}}{9}.

$$y=\frac{10\sqrt{5}}{\frac{50}{9}\times 9}$$

Divide \frac{10\sqrt{5}}{9} by \frac{50}{9} by multiplying \frac{10\sqrt{5}}{9} by the reciprocal of \frac{50}{9}.

$$y=\frac{\sqrt{5}}{5}$$

There are two solutions for y: \frac{11\sqrt{5}}{25} and \frac{\sqrt{5}}{5}. Substitute \frac{11\sqrt{5}}{25} for y in the equation x=-\frac{4}{3}y+\frac{2\sqrt{5}}{3} to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{4}{3}\times \frac{11\sqrt{5}}{25}+\frac{2\sqrt{5}}{3}$$

Multiply -\frac{4}{3} times \frac{11\sqrt{5}}{25}.

$$x=\frac{-4\times \frac{11\sqrt{5}}{25}+2\sqrt{5}}{3}$$

Now substitute \frac{\sqrt{5}}{5} for y in the equation x=-\frac{4}{3}y+\frac{2\sqrt{5}}{3} and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{4}{3}\times \frac{\sqrt{5}}{5}+\frac{2\sqrt{5}}{3}$$

Multiply -\frac{4}{3} times \frac{\sqrt{5}}{5}.

$$x=\frac{-4\times \frac{\sqrt{5}}{5}+2\sqrt{5}}{3}$$

The system is now solved.

$$x=\frac{-4\times \frac{11\sqrt{5}}{25}+2\sqrt{5}}{3},y=\frac{11\sqrt{5}}{25}\text{ or }x=\frac{-4\times \frac{\sqrt{5}}{5}+2\sqrt{5}}{3},y=\frac{\sqrt{5}}{5}$$