Solve {l}{1/3x+1/3y=1/4}{5/6x+frac{1}{4}=frac{2}{3}}

$$\left\{ \begin{array} { l } { \frac { 1 } { 3 x } + \frac { 1 } { 3 y } = \frac { 1 } { 4 } } \\ { \frac { 5 } { 6 x } + \frac { 1 } { 4 } = \frac { 2 } { 3 } } \end{array} \right.$$

Solve for x, y

$$x=2$$

$$y=4$$

Solution Steps

Consider the second equation. Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 12x, the least common multiple of 6x,4,3.

$$2\times 5+12x\times \frac{1}{4}=8x$$

Multiply 2 and 5 to get 10.

$$10+12x\times \frac{1}{4}=8x$$

Multiply 12 and \frac{1}{4} to get 3.

$$10+3x=8x$$

Subtract 8x from both sides.

$$10+3x-8x=0$$

Combine 3x and -8x to get -5x.

$$10-5x=0$$

Subtract 10 from both sides. Anything subtracted from zero gives its negation.

$$-5x=-10$$

Divide both sides by -5.

$$x=\frac{-10}{-5}$$

Divide -10 by -5 to get 2.

$$x=2$$

Consider the first equation. Insert the known values of variables into the equation.

$$\frac{1}{3\times 2}+\frac{1}{3y}=\frac{1}{4}$$

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 12y, the least common multiple of 3y,4.

$$2y+4=3y$$

Subtract 3y from both sides.

$$2y+4-3y=0$$

Combine 2y and -3y to get -y.

$$-y+4=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$-y=-4$$

Divide both sides by -1.

$$y=\frac{-4}{-1}$$

Fraction \frac{-4}{-1} can be simplified to 4 by removing the negative sign from both the numerator and the denominator.

$$y=4$$

The system is now solved.

$$x=2 y=4$$