Solve {l}{y=x/3+7/3}{y=sqrt{3}x-3sqrt{3}}

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

Solve for y, x

$$x = \frac{15 \sqrt{3} + 44}{13} \approx 5.383135547$$

$$y = \frac{5 \sqrt{3} + 45}{13} \approx 4.127711849$$

Steps Using Substitution

Consider the first equation. Since \frac{x}{3} and \frac{7}{3} have the same denominator, add them by adding their numerators.

$$y=\frac{x+7}{3}$$

Divide each term of x+7 by 3 to get \frac{1}{3}x+\frac{7}{3}.

$$y=\frac{1}{3}x+\frac{7}{3}$$

Substitute \frac{7+x}{3} for y in the other equation, y+\left(-\sqrt{3}\right)x=-3\sqrt{3}.

$$\frac{1}{3}x+\frac{7}{3}+\left(-\sqrt{3}\right)x=-3\sqrt{3}$$

Add \frac{x}{3} to -\sqrt{3}x.

$$\left(\frac{1}{3}-\sqrt{3}\right)x+\frac{7}{3}=-3\sqrt{3}$$

Subtract \frac{7}{3} from both sides of the equation.

$$\left(\frac{1}{3}-\sqrt{3}\right)x=-3\sqrt{3}-\frac{7}{3}$$

Divide both sides by \frac{1}{3}-\sqrt{3}.

$$x=\frac{15\sqrt{3}+44}{13}$$

Substitute \frac{44+15\sqrt{3}}{13} for x in y=\frac{1}{3}x+\frac{7}{3}. Because the resulting equation contains only one variable, you can solve for y directly.

$$y=\frac{1}{3}\times \frac{15\sqrt{3}+44}{13}+\frac{7}{3}$$

Multiply \frac{1}{3} times \frac{44+15\sqrt{3}}{13}.

$$y=\frac{5\sqrt{3}}{13}+\frac{44}{39}+\frac{7}{3}$$

Add \frac{7}{3} to \frac{44}{39}+\frac{5\sqrt{3}}{13}.

$$y=\frac{5\sqrt{3}+45}{13}$$

The system is now solved.

$$y=\frac{5\sqrt{3}+45}{13},x=\frac{15\sqrt{3}+44}{13}$$

Steps Using Elimination

Consider the first equation. Since \frac{x}{3} and \frac{7}{3} have the same denominator, add them by adding their numerators.

$$y=\frac{x+7}{3}$$

Divide each term of x+7 by 3 to get \frac{1}{3}x+\frac{7}{3}.

$$y=\frac{1}{3}x+\frac{7}{3}$$

Subtract \frac{1}{3}x from both sides.

$$y-\frac{1}{3}x=\frac{7}{3}$$

Consider the second equation. Subtract \sqrt{3}x from both sides.

$$y-\sqrt{3}x=-3\sqrt{3}$$

Reorder the terms.

$$-\sqrt{3}x+y=-3\sqrt{3}$$

In order to solve by elimination, coefficients of one of the variables must be the same in both equations so that the variable will cancel out when one equation is subtracted from the other.

$$y-\frac{1}{3}x=\frac{7}{3},y+\left(-\sqrt{3}\right)x=-3\sqrt{3}$$

Subtract y+\left(-\sqrt{3}\right)x=-3\sqrt{3} from y-\frac{1}{3}x=\frac{7}{3} by subtracting like terms on each side of the equal sign.

$$y-y-\frac{1}{3}x+\sqrt{3}x=\frac{7}{3}+3\sqrt{3}$$

Add y to -y. Terms y and -y cancel out, leaving an equation with only one variable that can be solved.

$$-\frac{1}{3}x+\sqrt{3}x=\frac{7}{3}+3\sqrt{3}$$

Add -\frac{x}{3} to \sqrt{3}x.

$$\left(\sqrt{3}-\frac{1}{3}\right)x=\frac{7}{3}+3\sqrt{3}$$

Add \frac{7}{3} to 3\sqrt{3}.

$$\left(\sqrt{3}-\frac{1}{3}\right)x=3\sqrt{3}+\frac{7}{3}$$

Divide both sides by -\frac{1}{3}+\sqrt{3}.

$$x=\frac{15\sqrt{3}+44}{13}$$

Substitute \frac{15\sqrt{3}+44}{13} for x in y+\left(-\sqrt{3}\right)x=-3\sqrt{3}. Because the resulting equation contains only one variable, you can solve for y directly.

$$y+\left(-\sqrt{3}\right)\times \frac{15\sqrt{3}+44}{13}=-3\sqrt{3}$$

Multiply -\sqrt{3} times \frac{15\sqrt{3}+44}{13}.

$$y+\frac{-44\sqrt{3}-45}{13}=-3\sqrt{3}$$

Subtract \frac{-45-44\sqrt{3}}{13} from both sides of the equation.

$$y=\frac{5\sqrt{3}+45}{13}$$

The system is now solved.

$$y=\frac{5\sqrt{3}+45}{13},x=\frac{15\sqrt{3}+44}{13}$$