Solve {l}{x^2+y^2=2}{x=6y}

$$\left. \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = 2 } \\ { x = 6 y } \end{array} \right.$$

Solve for x, y

$$x=-\frac{6\sqrt{74}}{37}\approx -1.394971665\text{, }y=-\frac{\sqrt{74}}{37}\approx -0.232495277$$

$$x=\frac{6\sqrt{74}}{37}\approx 1.394971665\text{, }y=\frac{\sqrt{74}}{37}\approx 0.232495277$$

Steps Using Substitution and Quadratic Formula

Consider the second equation. Subtract 6y from both sides.

$$x-6y=0$$

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.

$$x-6y=0,y^{2}+x^{2}=2$$

Solve x-6y=0 for x by isolating x on the left hand side of the equal sign.

$$x-6y=0$$

Subtract -6y from both sides of the equation.

$$x=6y$$

Substitute 6y for x in the other equation, y^{2}+x^{2}=2.

$$y^{2}+\left(6y\right)^{2}=2$$

Square 6y.

$$y^{2}+36y^{2}=2$$

Add y^{2} to 36y^{2}.

$$37y^{2}=2$$

Subtract 2 from both sides of the equation.

$$37y^{2}-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\times 6^{2} for a, 1\times 0\times 2\times 6 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{0±\sqrt{0^{2}-4\times 37\left(-2\right)}}{2\times 37}$$

Square 1\times 0\times 2\times 6.

$$y=\frac{0±\sqrt{-4\times 37\left(-2\right)}}{2\times 37}$$

Multiply -4 times 1+1\times 6^{2}.

$$y=\frac{0±\sqrt{-148\left(-2\right)}}{2\times 37}$$

Multiply -148 times -2.

$$y=\frac{0±\sqrt{296}}{2\times 37}$$

Take the square root of 296.

$$y=\frac{0±2\sqrt{74}}{2\times 37}$$

Multiply 2 times 1+1\times 6^{2}.

$$y=\frac{0±2\sqrt{74}}{74}$$

Now solve the equation y=\frac{0±2\sqrt{74}}{74} when ± is plus.

$$y=\frac{\sqrt{74}}{37}$$

Now solve the equation y=\frac{0±2\sqrt{74}}{74} when ± is minus.

$$y=-\frac{\sqrt{74}}{37}$$

There are two solutions for y: \frac{\sqrt{74}}{37} and -\frac{\sqrt{74}}{37}. Substitute \frac{\sqrt{74}}{37} for y in the equation x=6y to find the corresponding solution for x that satisfies both equations.

$$x=6\times \frac{\sqrt{74}}{37}$$

Now substitute -\frac{\sqrt{74}}{37} for y in the equation x=6y and solve to find the corresponding solution for x that satisfies both equations.

$$x=6\left(-\frac{\sqrt{74}}{37}\right)$$

The system is now solved.

$$x=6\times \frac{\sqrt{74}}{37},y=\frac{\sqrt{74}}{37}\text{ or }x=6\left(-\frac{\sqrt{74}}{37}\right),y=-\frac{\sqrt{74}}{37}$$