Solve {l}{x^2+y^2=(57.15)^2}{x/y=10/16}

$$\left\{ \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = ( 57.15 ) ^ { 2 } } \\ { x / y = \frac { 10 } { 16 } } \end{array} \right.$$

Solve for x, y

$$x=-\frac{1143\sqrt{89}}{356}\approx -30.289439421\text{, }y=-\frac{2286\sqrt{89}}{445}\approx -48.463103074$$

$$x=\frac{1143\sqrt{89}}{356}\approx 30.289439421\text{, }y=\frac{2286\sqrt{89}}{445}\approx 48.463103074$$

Steps Using Substitution and Quadratic Formula

Consider the first equation. Calculate 57.15 to the power of 2 and get 3266.1225.

$$x^{2}+y^{2}=3266.1225$$

Consider the second equation. Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 16y, the least common multiple of y,16.

$$16x=y\times 10$$

Subtract y\times 10 from both sides.

$$16x-y\times 10=0$$

Multiply -1 and 10 to get -10.

$$16x-10y=0$$

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.

$$16x-10y=0,y^{2}+x^{2}=3266.1225$$

Solve 16x-10y=0 for x by isolating x on the left hand side of the equal sign.

$$16x-10y=0$$

Subtract -10y from both sides of the equation.

$$16x=10y$$

Divide both sides by 16.

$$x=\frac{5}{8}y$$

Substitute \frac{5}{8}y for x in the other equation, y^{2}+x^{2}=3266.1225.

$$y^{2}+\left(\frac{5}{8}y\right)^{2}=3266.1225$$

Square \frac{5}{8}y.

$$y^{2}+\frac{25}{64}y^{2}=3266.1225$$

Add y^{2} to \frac{25}{64}y^{2}.

$$\frac{89}{64}y^{2}=3266.1225$$

Subtract 3266.1225 from both sides of the equation.

$$\frac{89}{64}y^{2}-3266.1225=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\times \left(\frac{5}{8}\right)^{2} for a, 1\times 0\times \frac{5}{8}\times 2 for b, and -3266.1225 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{0±\sqrt{0^{2}-4\times \frac{89}{64}\left(-3266.1225\right)}}{2\times \frac{89}{64}}$$

Square 1\times 0\times \frac{5}{8}\times 2.

$$y=\frac{0±\sqrt{-4\times \frac{89}{64}\left(-3266.1225\right)}}{2\times \frac{89}{64}}$$

Multiply -4 times 1+1\times \left(\frac{5}{8}\right)^{2}.

$$y=\frac{0±\sqrt{-\frac{89}{16}\left(-3266.1225\right)}}{2\times \frac{89}{64}}$$

Multiply -\frac{89}{16} times -3266.1225 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{0±\sqrt{\frac{116273961}{6400}}}{2\times \frac{89}{64}}$$

Take the square root of \frac{116273961}{6400}.

$$y=\frac{0±\frac{1143\sqrt{89}}{80}}{2\times \frac{89}{64}}$$

Multiply 2 times 1+1\times \left(\frac{5}{8}\right)^{2}.

$$y=\frac{0±\frac{1143\sqrt{89}}{80}}{\frac{89}{32}}$$

Now solve the equation y=\frac{0±\frac{1143\sqrt{89}}{80}}{\frac{89}{32}} when ± is plus.

$$y=\frac{2286\sqrt{89}}{445}$$

Now solve the equation y=\frac{0±\frac{1143\sqrt{89}}{80}}{\frac{89}{32}} when ± is minus.

$$y=-\frac{2286\sqrt{89}}{445}$$

There are two solutions for y: \frac{2286\sqrt{89}}{445} and -\frac{2286\sqrt{89}}{445}. Substitute \frac{2286\sqrt{89}}{445} for y in the equation x=\frac{5}{8}y to find the corresponding solution for x that satisfies both equations.

$$x=\frac{5}{8}\times \frac{2286\sqrt{89}}{445}$$

Multiply \frac{5}{8} times \frac{2286\sqrt{89}}{445}.

$$x=\frac{5\times \frac{2286\sqrt{89}}{445}}{8}$$

Now substitute -\frac{2286\sqrt{89}}{445} for y in the equation x=\frac{5}{8}y and solve to find the corresponding solution for x that satisfies both equations.

$$x=\frac{5}{8}\left(-\frac{2286\sqrt{89}}{445}\right)$$

Multiply \frac{5}{8} times -\frac{2286\sqrt{89}}{445}.

$$x=\frac{5\left(-\frac{2286\sqrt{89}}{445}\right)}{8}$$

The system is now solved.

$$x=\frac{5\times \frac{2286\sqrt{89}}{445}}{8},y=\frac{2286\sqrt{89}}{445}\text{ or }x=\frac{5\left(-\frac{2286\sqrt{89}}{445}\right)}{8},y=-\frac{2286\sqrt{89}}{445}$$