Solve {l}{sqrt{3}x-5y=0}{3sqrt{3}x-y-2asqrt{3}=0}

$$\left\{ \begin{array} { l } { \sqrt { 3 } x - 5 y = 0 } \\ { 3 \sqrt { 3 } x - y - 2 a \sqrt { 3 } = 0 } \end{array} \right.$$

Solve for x, y

$$x=\frac{5a}{7}$$

$$y=\frac{\sqrt{3}a}{7}$$

Steps Using Substitution

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.

$$\sqrt{3}x-5y=0,3\sqrt{3}x-y-2\sqrt{3}a=0$$

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

$$\sqrt{3}x-5y=0$$

Add 5y to both sides of the equation.

$$\sqrt{3}x=5y$$

Divide both sides by \sqrt{3}.

$$x=\frac{\sqrt{3}}{3}\times 5y$$

Multiply \frac{\sqrt{3}}{3} times 5y.

$$x=\frac{5\sqrt{3}}{3}y$$

Substitute \frac{5\sqrt{3}y}{3} for x in the other equation, 3\sqrt{3}x-y-2\sqrt{3}a=0.

$$3\sqrt{3}\times \frac{5\sqrt{3}}{3}y-y-2\sqrt{3}a=0$$

Multiply 3\sqrt{3} times \frac{5\sqrt{3}y}{3}.

$$15y-y-2\sqrt{3}a=0$$

Add 15y to -y.

$$14y-2\sqrt{3}a=0$$

Add 2a\sqrt{3} to both sides of the equation.

$$14y=2\sqrt{3}a$$

Divide both sides by 14.

$$y=\frac{\sqrt{3}a}{7}$$

Substitute \frac{a\sqrt{3}}{7} for y in x=\frac{5\sqrt{3}}{3}y. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{5\sqrt{3}}{3}\times \frac{\sqrt{3}a}{7}$$

Multiply \frac{5\sqrt{3}}{3} times \frac{a\sqrt{3}}{7}.

$$x=\frac{5a}{7}$$

The system is now solved.

$$x=\frac{5a}{7},y=\frac{\sqrt{3}a}{7}$$

Steps Using Elimination

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.

$$\sqrt{3}x-5y=0,3\sqrt{3}x-y-2\sqrt{3}a=0$$

To make \sqrt{3}x and 3x\sqrt{3} equal, multiply all terms on each side of the first equation by 3\sqrt{3} and all terms on each side of the second by \sqrt{3}.

$$3\sqrt{3}\sqrt{3}x+3\sqrt{3}\left(-5\right)y=0,\sqrt{3}\times 3\sqrt{3}x+\sqrt{3}\left(-1\right)y+\sqrt{3}\left(-2\sqrt{3}a\right)=0$$

Simplify.

$$9x+\left(-15\sqrt{3}\right)y=0,9x+\left(-\sqrt{3}\right)y-6a=0$$

Subtract 9x+\left(-\sqrt{3}\right)y-6a=0 from 9x+\left(-15\sqrt{3}\right)y=0 by subtracting like terms on each side of the equal sign.

$$9x-9x+\left(-15\sqrt{3}\right)y+\sqrt{3}y+6a=0$$

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

$$\left(-15\sqrt{3}\right)y+\sqrt{3}y+6a=0$$

Add -15\sqrt{3}y to \sqrt{3}y.

$$\left(-14\sqrt{3}\right)y+6a=0$$

Subtract 6a from both sides of the equation.

$$\left(-14\sqrt{3}\right)y=-6a$$

Divide both sides by -14\sqrt{3}.

$$y=\frac{\sqrt{3}a}{7}$$

Substitute \frac{a\sqrt{3}}{7} for y in 3\sqrt{3}x-y-2\sqrt{3}a=0. Because the resulting equation contains only one variable, you can solve for x directly.

$$3\sqrt{3}x-\frac{\sqrt{3}a}{7}-2\sqrt{3}a=0$$

Add -\frac{a\sqrt{3}}{7} to -2a\sqrt{3}.

$$3\sqrt{3}x-\frac{15\sqrt{3}a}{7}=0$$

Add \frac{15a\sqrt{3}}{7} to both sides of the equation.

$$3\sqrt{3}x=\frac{15\sqrt{3}a}{7}$$

Divide both sides by 3\sqrt{3}.

$$x=\frac{5a}{7}$$

The system is now solved.

$$x=\frac{5a}{7},y=\frac{\sqrt{3}a}{7}$$