Solve {l}{frac{sqrt{2}}{2}y+7sqrt{2}=x}{frac{sqrt{2}}{2}y=7sqrt{6}}

$$\left. \begin{array} { l } { \frac { \sqrt { 2 } } { 2 } y + 7 \sqrt { 2 } = x } \\ { \frac { \sqrt { 2 } } { 2 } y = 7 \sqrt { 6 } } \end{array} \right.$$

Solve for y, x

$$x=7\left(\sqrt{2}+\sqrt{6}\right)\approx 27.045923136$$

$$y=14\sqrt{3}\approx 24.248711306$$

Steps Using Substitution

Consider the second equation. Multiply both sides of the equation by 2.

$$\sqrt{2}y=14\sqrt{6}$$

Consider the first equation. Express \frac{\sqrt{2}}{2}y as a single fraction.

$$\frac{\sqrt{2}y}{2}+7\sqrt{2}=x$$

Subtract x from both sides.

$$\frac{\sqrt{2}y}{2}+7\sqrt{2}-x=0$$

Subtract 7\sqrt{2} from both sides. Anything subtracted from zero gives its negation.

$$\frac{\sqrt{2}y}{2}-x=-7\sqrt{2}$$

Multiply both sides of the equation by 2.

$$\sqrt{2}y-2x=-14\sqrt{2}$$

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.

$$\sqrt{2}y=14\sqrt{6},\sqrt{2}y-2x=-14\sqrt{2}$$

Pick one of the two equations which is more simple to solve for y by isolating y on the left hand side of the equal sign.

$$\sqrt{2}y=14\sqrt{6}$$

Divide both sides by \sqrt{2}.

$$y=14\sqrt{3}$$

Substitute 14\sqrt{3} for y in the other equation, \sqrt{2}y-2x=-14\sqrt{2}.

$$\sqrt{2}\times 14\sqrt{3}-2x=-14\sqrt{2}$$

Multiply \sqrt{2} times 14\sqrt{3}.

$$14\sqrt{6}-2x=-14\sqrt{2}$$

Subtract 14\sqrt{6} from both sides of the equation.

$$-2x=-14\sqrt{2}-14\sqrt{6}$$

Divide both sides by -2.

$$x=7\sqrt{2}+7\sqrt{6}$$

The system is now solved.

$$y=14\sqrt{3},x=7\sqrt{2}+7\sqrt{6}$$