Solve {l}{y=(x-sqrt{2})}{x^2+3y^2=3}

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

Solve for y, x

$$x=\frac{3\sqrt{2}-\sqrt{6}}{4}\approx 0.448287736\text{, }y=\frac{-\sqrt{2}-\sqrt{6}}{4}\approx -0.965925826$$

$$x=\frac{\sqrt{6}+3\sqrt{2}}{4}\approx 1.673032607\text{, }y=\frac{\sqrt{6}-\sqrt{2}}{4}\approx 0.258819045$$

Steps Using Substitution and Quadratic Formula

Consider the first equation. Subtract x from both sides.

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

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.

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

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

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

Subtract -x from both sides of the equation.

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

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

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

Square x-\sqrt{2}.

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

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

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

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

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

Subtract 3 from both sides of the equation.

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

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

$$x=\frac{-6\left(-\sqrt{2}\right)±\sqrt{\left(6\left(-\sqrt{2}\right)\right)^{2}-4\times 4\times 3}}{2\times 4}$$

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

$$x=\frac{-6\left(-\sqrt{2}\right)±\sqrt{72-4\times 4\times 3}}{2\times 4}$$

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

$$x=\frac{-6\left(-\sqrt{2}\right)±\sqrt{72-16\times 3}}{2\times 4}$$

Multiply -16 times 3.

$$x=\frac{-6\left(-\sqrt{2}\right)±\sqrt{72-48}}{2\times 4}$$

Add 72 to -48.

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

Take the square root of 24.

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

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

$$x=\frac{6\sqrt{2}±2\sqrt{6}}{2\times 4}$$

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

$$x=\frac{6\sqrt{2}±2\sqrt{6}}{8}$$

Now solve the equation x=\frac{6\sqrt{2}±2\sqrt{6}}{8} when ± is plus. Add 6\sqrt{2} to 2\sqrt{6}.

$$x=\frac{2\sqrt{6}+6\sqrt{2}}{8}$$

Divide 6\sqrt{2}+2\sqrt{6} by 8.

$$x=\frac{\sqrt{6}+3\sqrt{2}}{4}$$

Now solve the equation x=\frac{6\sqrt{2}±2\sqrt{6}}{8} when ± is minus. Subtract 2\sqrt{6} from 6\sqrt{2}.

$$x=\frac{6\sqrt{2}-2\sqrt{6}}{8}$$

Divide 6\sqrt{2}-2\sqrt{6} by 8.

$$x=\frac{3\sqrt{2}-\sqrt{6}}{4}$$

There are two solutions for x: \frac{3\sqrt{2}+\sqrt{6}}{4} and \frac{3\sqrt{2}-\sqrt{6}}{4}. Substitute \frac{3\sqrt{2}+\sqrt{6}}{4} for x in the equation y=x-\sqrt{2} to find the corresponding solution for y that satisfies both equations.

$$y=\frac{\sqrt{6}+3\sqrt{2}}{4}-\sqrt{2}$$

Now substitute \frac{3\sqrt{2}-\sqrt{6}}{4} for x in the equation y=x-\sqrt{2} and solve to find the corresponding solution for y that satisfies both equations.

$$y=\frac{3\sqrt{2}-\sqrt{6}}{4}-\sqrt{2}$$

The system is now solved.

$$y=\frac{\sqrt{6}+3\sqrt{2}}{4}-\sqrt{2},x=\frac{\sqrt{6}+3\sqrt{2}}{4}\text{ or }y=\frac{3\sqrt{2}-\sqrt{6}}{4}-\sqrt{2},x=\frac{3\sqrt{2}-\sqrt{6}}{4}$$