Solve {l}{y=-3sqrt{3}x+6sqrt{3}}{y=1/3sqrt{3}x}

$$\left\{ \begin{array} { l } { y = - 3 \sqrt { 3 } x + 6 \sqrt { 3 } } \\ { y = \frac { 1 } { 3 \sqrt { 3 } } x } \end{array} \right.$$

Solve for y, x

$$x = \frac{27}{14} = 1\frac{13}{14} \approx 1.928571429$$

$$y=\frac{3\sqrt{3}}{14}\approx 0.371153744$$

Steps Using Substitution

Consider the first equation. Add 3\sqrt{3}x to both sides.

$$y+3\sqrt{3}x=6\sqrt{3}$$

Consider the second equation. Rationalize the denominator of \frac{1}{3\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$y=\frac{\sqrt{3}}{3\left(\sqrt{3}\right)^{2}}x$$

The square of \sqrt{3} is 3.

$$y=\frac{\sqrt{3}}{3\times 3}x$$

Multiply 3 and 3 to get 9.

$$y=\frac{\sqrt{3}}{9}x$$

Express \frac{\sqrt{3}}{9}x as a single fraction.

$$y=\frac{\sqrt{3}x}{9}$$

Subtract \frac{\sqrt{3}x}{9} from both sides.

$$y-\frac{\sqrt{3}x}{9}=0$$

Multiply both sides of the equation by 9.

$$9y-\sqrt{3}x=0$$

Reorder the terms.

$$-\sqrt{3}x+9y=0$$

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.

$$y+3\sqrt{3}x=6\sqrt{3},9y+\left(-\sqrt{3}\right)x=0$$

Choose one of the equations and solve it for y by isolating y on the left hand side of the equal sign.

$$y+3\sqrt{3}x=6\sqrt{3}$$

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

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

Substitute 3\left(2-x\right)\sqrt{3} for y in the other equation, 9y+\left(-\sqrt{3}\right)x=0.

$$9\left(\left(-3\sqrt{3}\right)x+6\sqrt{3}\right)+\left(-\sqrt{3}\right)x=0$$

Multiply 9 times 3\left(2-x\right)\sqrt{3}.

$$\left(-27\sqrt{3}\right)x+54\sqrt{3}+\left(-\sqrt{3}\right)x=0$$

Add -27\sqrt{3}x to -\sqrt{3}x.

$$\left(-28\sqrt{3}\right)x+54\sqrt{3}=0$$

Subtract 54\sqrt{3} from both sides of the equation.

$$\left(-28\sqrt{3}\right)x=-54\sqrt{3}$$

Divide both sides by -28\sqrt{3}.

$$x=\frac{27}{14}$$

Substitute \frac{27}{14} for x in y=\left(-3\sqrt{3}\right)x+6\sqrt{3}. Because the resulting equation contains only one variable, you can solve for y directly.

$$y=\left(-3\sqrt{3}\right)\times \frac{27}{14}+6\sqrt{3}$$

Multiply -3\sqrt{3} times \frac{27}{14}.

$$y=-\frac{81\sqrt{3}}{14}+6\sqrt{3}$$

Add 6\sqrt{3} to -\frac{81\sqrt{3}}{14}.

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

The system is now solved.

$$y=\frac{3\sqrt{3}}{14},x=\frac{27}{14}$$

Steps Using Elimination

Consider the first equation. Add 3\sqrt{3}x to both sides.

$$y+3\sqrt{3}x=6\sqrt{3}$$

Consider the second equation. Rationalize the denominator of \frac{1}{3\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$y=\frac{\sqrt{3}}{3\left(\sqrt{3}\right)^{2}}x$$

The square of \sqrt{3} is 3.

$$y=\frac{\sqrt{3}}{3\times 3}x$$

Multiply 3 and 3 to get 9.

$$y=\frac{\sqrt{3}}{9}x$$

Express \frac{\sqrt{3}}{9}x as a single fraction.

$$y=\frac{\sqrt{3}x}{9}$$

Subtract \frac{\sqrt{3}x}{9} from both sides.

$$y-\frac{\sqrt{3}x}{9}=0$$

Multiply both sides of the equation by 9.

$$9y-\sqrt{3}x=0$$

Reorder the terms.

$$-\sqrt{3}x+9y=0$$

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.

$$y+3\sqrt{3}x=6\sqrt{3},9y+\left(-\sqrt{3}\right)x=0$$

To make y and 9y equal, multiply all terms on each side of the first equation by 9 and all terms on each side of the second by 1.

$$9y+9\times 3\sqrt{3}x=9\times 6\sqrt{3},9y+\left(-\sqrt{3}\right)x=0$$

Simplify.

$$9y+27\sqrt{3}x=54\sqrt{3},9y+\left(-\sqrt{3}\right)x=0$$

Subtract 9y+\left(-\sqrt{3}\right)x=0 from 9y+27\sqrt{3}x=54\sqrt{3} by subtracting like terms on each side of the equal sign.

$$9y-9y+27\sqrt{3}x+\sqrt{3}x=54\sqrt{3}$$

Add 9y to -9y. Terms 9y and -9y cancel out, leaving an equation with only one variable that can be solved.

$$27\sqrt{3}x+\sqrt{3}x=54\sqrt{3}$$

Add 27x\sqrt{3} to \sqrt{3}x.

$$28\sqrt{3}x=54\sqrt{3}$$

Divide both sides by 28\sqrt{3}.

$$x=\frac{27}{14}$$

Substitute \frac{27}{14} for x in 9y+\left(-\sqrt{3}\right)x=0. Because the resulting equation contains only one variable, you can solve for y directly.

$$9y+\left(-\sqrt{3}\right)\times \frac{27}{14}=0$$

Multiply -\sqrt{3} times \frac{27}{14}.

$$9y-\frac{27\sqrt{3}}{14}=0$$

Add \frac{27\sqrt{3}}{14} to both sides of the equation.

$$9y=\frac{27\sqrt{3}}{14}$$

Divide both sides by 9.

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

The system is now solved.

$$y=\frac{3\sqrt{3}}{14},x=\frac{27}{14}$$