Solve {l}{x^2+y^2=10}{x=y}

$$\left. \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = 10 } \\ { x = y } \end{array} \right.$$

Solve for x, y

$$x=-\sqrt{5}\approx -2.236067977\text{, }y=-\sqrt{5}\approx -2.236067977$$

$$x=\sqrt{5}\approx 2.236067977\text{, }y=\sqrt{5}\approx 2.236067977$$

Steps Using Substitution and Quadratic Formula

Consider the second equation. Subtract y from both sides.

$$x-y=0$$

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.

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

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, y^{2}+x^{2}=10.

$$y^{2}+y^{2}=10$$

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

$$2y^{2}=10$$

Subtract 10 from both sides of the equation.

$$2y^{2}-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\times 1^{2} for a, 1\times 0\times 1\times 2 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{0±\sqrt{0^{2}-4\times 2\left(-10\right)}}{2\times 2}$$

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

$$y=\frac{0±\sqrt{-4\times 2\left(-10\right)}}{2\times 2}$$

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

$$y=\frac{0±\sqrt{-8\left(-10\right)}}{2\times 2}$$

Multiply -8 times -10.

$$y=\frac{0±\sqrt{80}}{2\times 2}$$

Take the square root of 80.

$$y=\frac{0±4\sqrt{5}}{2\times 2}$$

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

$$y=\frac{0±4\sqrt{5}}{4}$$

Now solve the equation y=\frac{0±4\sqrt{5}}{4} when ± is plus.

$$y=\sqrt{5}$$

Now solve the equation y=\frac{0±4\sqrt{5}}{4} when ± is minus.

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

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

$$x=\sqrt{5}$$

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

$$x=-\sqrt{5}$$

The system is now solved.

$$x=\sqrt{5},y=\sqrt{5}\text{ or }x=-\sqrt{5},y=-\sqrt{5}$$