Solve {l}{x-y=0}{3x^2+3y^2=24}

$$\left\{ \begin{array} { l } { x - y = 0 } \\ { 3 x ^ { 2 } + 3 y ^ { 2 } = 24 } \end{array} \right.$$

Solve for x, y

$$x=-2\text{, }y=-2$$

$$x=2\text{, }y=2$$

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.

$$x-y=0,3y^{2}+3x^{2}=24$$

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

$$x-y=0$$

Subtract -y from both sides of the equation.

$$x=y$$

Substitute y for x in the other equation, 3y^{2}+3x^{2}=24.

$$3y^{2}+3y^{2}=24$$

Add 3y^{2} to 3y^{2}.

$$6y^{2}=24$$

Subtract 24 from both sides of the equation.

$$6y^{2}-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3+3\times 1^{2} for a, 3\times 0\times 1\times 2 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{0±\sqrt{0^{2}-4\times 6\left(-24\right)}}{2\times 6}$$

Square 3\times 0\times 1\times 2.

$$y=\frac{0±\sqrt{-4\times 6\left(-24\right)}}{2\times 6}$$

Multiply -4 times 3+3\times 1^{2}.

$$y=\frac{0±\sqrt{-24\left(-24\right)}}{2\times 6}$$

Multiply -24 times -24.

$$y=\frac{0±\sqrt{576}}{2\times 6}$$

Take the square root of 576.

$$y=\frac{0±24}{2\times 6}$$

Multiply 2 times 3+3\times 1^{2}.

$$y=\frac{0±24}{12}$$

Now solve the equation y=\frac{0±24}{12} when ± is plus. Divide 24 by 12.

$$y=2$$

Now solve the equation y=\frac{0±24}{12} when ± is minus. Divide -24 by 12.

$$y=-2$$

There are two solutions for y: 2 and -2. Substitute 2 for y in the equation x=y to find the corresponding solution for x that satisfies both equations.

$$x=2$$

Now substitute -2 for y in the equation x=y and solve to find the corresponding solution for x that satisfies both equations.

$$x=-2$$

The system is now solved.

$$x=2,y=2\text{ or }x=-2,y=-2$$