Solve {l}{x+y=sqrt{2}}{3x^2+y^2=sqrt{7}}

$$\left. \begin{array} { l } { x + y = \sqrt { 2 } } \\ { 3 x ^ { 2 } + y ^ { 2 } = \sqrt { 7 } } \end{array} \right.$$

Solve for x, y

$$x=\frac{\sqrt{4\sqrt{7}-6}+\sqrt{2}}{4}\approx 0.888752258\text{, }y=\frac{-\sqrt{4\sqrt{7}-6}+3\sqrt{2}}{4}\approx 0.525461304$$

$$x=\frac{-\sqrt{4\sqrt{7}-6}+\sqrt{2}}{4}\approx -0.181645477\text{, }y=\frac{\sqrt{4\sqrt{7}-6}+3\sqrt{2}}{4}\approx 1.595859039$$

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.

$$x+y=\sqrt{2},y^{2}+3x^{2}=\sqrt{7}$$

Solve x+y=\sqrt{2} for x by isolating x on the left hand side of the equal sign.

$$x+y=\sqrt{2}$$

Subtract y from both sides of the equation.

$$x=-y+\sqrt{2}$$

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

$$y^{2}+3\left(-y+\sqrt{2}\right)^{2}=\sqrt{7}$$

Square -y+\sqrt{2}.

$$y^{2}+3\left(y^{2}+\left(-2\sqrt{2}\right)y+\left(\sqrt{2}\right)^{2}\right)=\sqrt{7}$$

Multiply 3 times y^{2}+\left(-2\sqrt{2}\right)y+\left(\sqrt{2}\right)^{2}.

$$y^{2}+3y^{2}+\left(-6\sqrt{2}\right)y+3\left(\sqrt{2}\right)^{2}=\sqrt{7}$$

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

$$4y^{2}+\left(-6\sqrt{2}\right)y+3\left(\sqrt{2}\right)^{2}=\sqrt{7}$$

Subtract \sqrt{7} from both sides of the equation.

$$4y^{2}+\left(-6\sqrt{2}\right)y+3\left(\sqrt{2}\right)^{2}-\sqrt{7}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+3\left(-1\right)^{2} for a, 3\left(-1\right)\times 2\sqrt{2} for b, and 6-\sqrt{7} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-6\sqrt{2}\right)±\sqrt{\left(-6\sqrt{2}\right)^{2}-4\times 4\left(6-\sqrt{7}\right)}}{2\times 4}$$

Square 3\left(-1\right)\times 2\sqrt{2}.

$$y=\frac{-\left(-6\sqrt{2}\right)±\sqrt{72-4\times 4\left(6-\sqrt{7}\right)}}{2\times 4}$$

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

$$y=\frac{-\left(-6\sqrt{2}\right)±\sqrt{72-16\left(6-\sqrt{7}\right)}}{2\times 4}$$

Multiply -16 times 6-\sqrt{7}.

$$y=\frac{-\left(-6\sqrt{2}\right)±\sqrt{72+16\sqrt{7}-96}}{2\times 4}$$

Add 72 to -96+16\sqrt{7}.

$$y=\frac{-\left(-6\sqrt{2}\right)±\sqrt{16\sqrt{7}-24}}{2\times 4}$$

Take the square root of -24+16\sqrt{7}.

$$y=\frac{-\left(-6\sqrt{2}\right)±2\sqrt{4\sqrt{7}-6}}{2\times 4}$$

The opposite of 3\left(-1\right)\times 2\sqrt{2} is 6\sqrt{2}.

$$y=\frac{6\sqrt{2}±2\sqrt{4\sqrt{7}-6}}{2\times 4}$$

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

$$y=\frac{6\sqrt{2}±2\sqrt{4\sqrt{7}-6}}{8}$$

Now solve the equation y=\frac{6\sqrt{2}±2\sqrt{4\sqrt{7}-6}}{8} when ± is plus. Add 6\sqrt{2} to 2\sqrt{-6+4\sqrt{7}}.

$$y=\frac{2\sqrt{2}\left(\sqrt{2\sqrt{7}-3}+3\right)}{8}$$

Divide 2\left(3+\sqrt{-3+2\sqrt{7}}\right)\sqrt{2} by 8.

$$y=\frac{\sqrt{2}\left(\sqrt{2\sqrt{7}-3}+3\right)}{4}$$

Now solve the equation y=\frac{6\sqrt{2}±2\sqrt{4\sqrt{7}-6}}{8} when ± is minus. Subtract 2\sqrt{-6+4\sqrt{7}} from 6\sqrt{2}.

$$y=\frac{2\sqrt{2}\left(-\sqrt{2\sqrt{7}-3}+3\right)}{8}$$

Divide 2\left(3-\sqrt{-3+2\sqrt{7}}\right)\sqrt{2} by 8.

$$y=\frac{\sqrt{2}\left(-\sqrt{2\sqrt{7}-3}+3\right)}{4}$$

There are two solutions for y: \frac{\left(3+\sqrt{-3+2\sqrt{7}}\right)\sqrt{2}}{4} and \frac{\left(3-\sqrt{-3+2\sqrt{7}}\right)\sqrt{2}}{4}. Substitute \frac{\left(3+\sqrt{-3+2\sqrt{7}}\right)\sqrt{2}}{4} for y in the equation x=-y+\sqrt{2} to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{\sqrt{2}\left(\sqrt{2\sqrt{7}-3}+3\right)}{4}+\sqrt{2}$$

Now substitute \frac{\left(3-\sqrt{-3+2\sqrt{7}}\right)\sqrt{2}}{4} for y in the equation x=-y+\sqrt{2} and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{\sqrt{2}\left(-\sqrt{2\sqrt{7}-3}+3\right)}{4}+\sqrt{2}$$

The system is now solved.

$$x=-\frac{\sqrt{2}\left(\sqrt{2\sqrt{7}-3}+3\right)}{4}+\sqrt{2},y=\frac{\sqrt{2}\left(\sqrt{2\sqrt{7}-3}+3\right)}{4}\text{ or }x=-\frac{\sqrt{2}\left(-\sqrt{2\sqrt{7}-3}+3\right)}{4}+\sqrt{2},y=\frac{\sqrt{2}\left(-\sqrt{2\sqrt{7}-3}+3\right)}{4}$$