Solve {l}{x^2+y^2=30}{x:y=21:9}

$$\left. \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = 30 } \\ { x : y = 21 : 9 } \end{array} \right.$$

Solve for x, y

$$x=-\frac{7\sqrt{435}}{29}\approx -5.034364666\text{, }y=-\frac{3\sqrt{435}}{29}\approx -2.157584857$$

$$x=\frac{7\sqrt{435}}{29}\approx 5.034364666\text{, }y=\frac{3\sqrt{435}}{29}\approx 2.157584857$$

Steps Using Substitution and Quadratic Formula

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 9y, the least common multiple of y,9.

$$9x=y\times 21$$

Subtract y\times 21 from both sides.

$$9x-y\times 21=0$$

Multiply -1 and 21 to get -21.

$$9x-21y=0$$

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.

$$9x-21y=0,y^{2}+x^{2}=30$$

Solve 9x-21y=0 for x by isolating x on the left hand side of the equal sign.

$$9x-21y=0$$

Subtract -21y from both sides of the equation.

$$9x=21y$$

Divide both sides by 9.

$$x=\frac{7}{3}y$$

Substitute \frac{7}{3}y for x in the other equation, y^{2}+x^{2}=30.

$$y^{2}+\left(\frac{7}{3}y\right)^{2}=30$$

Square \frac{7}{3}y.

$$y^{2}+\frac{49}{9}y^{2}=30$$

Add y^{2} to \frac{49}{9}y^{2}.

$$\frac{58}{9}y^{2}=30$$

Subtract 30 from both sides of the equation.

$$\frac{58}{9}y^{2}-30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\times \left(\frac{7}{3}\right)^{2} for a, 1\times 0\times 2\times \frac{7}{3} for b, and -30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{0±\sqrt{0^{2}-4\times \frac{58}{9}\left(-30\right)}}{2\times \frac{58}{9}}$$

Square 1\times 0\times 2\times \frac{7}{3}.

$$y=\frac{0±\sqrt{-4\times \frac{58}{9}\left(-30\right)}}{2\times \frac{58}{9}}$$

Multiply -4 times 1+1\times \left(\frac{7}{3}\right)^{2}.

$$y=\frac{0±\sqrt{-\frac{232}{9}\left(-30\right)}}{2\times \frac{58}{9}}$$

Multiply -\frac{232}{9} times -30.

$$y=\frac{0±\sqrt{\frac{2320}{3}}}{2\times \frac{58}{9}}$$

Take the square root of \frac{2320}{3}.

$$y=\frac{0±\frac{4\sqrt{435}}{3}}{2\times \frac{58}{9}}$$

Multiply 2 times 1+1\times \left(\frac{7}{3}\right)^{2}.

$$y=\frac{0±\frac{4\sqrt{435}}{3}}{\frac{116}{9}}$$

Now solve the equation y=\frac{0±\frac{4\sqrt{435}}{3}}{\frac{116}{9}} when ± is plus.

$$y=\frac{3\sqrt{435}}{29}$$

Now solve the equation y=\frac{0±\frac{4\sqrt{435}}{3}}{\frac{116}{9}} when ± is minus.

$$y=-\frac{3\sqrt{435}}{29}$$

There are two solutions for y: \frac{3\sqrt{435}}{29} and -\frac{3\sqrt{435}}{29}. Substitute \frac{3\sqrt{435}}{29} for y in the equation x=\frac{7}{3}y to find the corresponding solution for x that satisfies both equations.

$$x=\frac{7}{3}\times \frac{3\sqrt{435}}{29}$$

Multiply \frac{7}{3} times \frac{3\sqrt{435}}{29}.

$$x=\frac{7\times \frac{3\sqrt{435}}{29}}{3}$$

Now substitute -\frac{3\sqrt{435}}{29} for y in the equation x=\frac{7}{3}y and solve to find the corresponding solution for x that satisfies both equations.

$$x=\frac{7}{3}\left(-\frac{3\sqrt{435}}{29}\right)$$

Multiply \frac{7}{3} times -\frac{3\sqrt{435}}{29}.

$$x=\frac{7\left(-\frac{3\sqrt{435}}{29}\right)}{3}$$

The system is now solved.

$$x=\frac{7\times \frac{3\sqrt{435}}{29}}{3},y=\frac{3\sqrt{435}}{29}\text{ or }x=\frac{7\left(-\frac{3\sqrt{435}}{29}\right)}{3},y=-\frac{3\sqrt{435}}{29}$$