Solve {l}{2:x/4+7/3=54}{4:x/4+y/5=42}

$$\left. \begin{array} { l } { 2 : \frac { x } { 4 } + \frac { 7 } { 3 } = 54 } \\ { 4 : \frac { x } { 4 } + \frac { y } { 5 } = 42 } \end{array} \right.$$

Solve for x, y

$$x=\frac{24}{155}\approx 0.15483871$$

$$y = -\frac{920}{3} = -306\frac{2}{3} \approx -306.666666667$$

Short Solution Steps

Consider the first equation. Divide 2 by \frac{x}{4} by multiplying 2 by the reciprocal of \frac{x}{4}.

$$\frac{2\times 4}{x}+\frac{7}{3}=54$$

Multiply 2 and 4 to get 8.

$$\frac{8}{x}+\frac{7}{3}=54$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and 3 is 3x. Multiply \frac{8}{x} times \frac{3}{3}. Multiply \frac{7}{3} times \frac{x}{x}.

$$\frac{8\times 3}{3x}+\frac{7x}{3x}=54$$

Since \frac{8\times 3}{3x} and \frac{7x}{3x} have the same denominator, add them by adding their numerators.

$$\frac{8\times 3+7x}{3x}=54$$

Do the multiplications in 8\times 3+7x.

$$\frac{24+7x}{3x}=54$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3x.

$$24+7x=162x$$

Subtract 162x from both sides.

$$24+7x-162x=0$$

Combine 7x and -162x to get -155x.

$$24-155x=0$$

Subtract 24 from both sides. Anything subtracted from zero gives its negation.

$$-155x=-24$$

Divide both sides by -155.

$$x=\frac{-24}{-155}$$

Fraction \frac{-24}{-155} can be simplified to \frac{24}{155} by removing the negative sign from both the numerator and the denominator.

$$x=\frac{24}{155}$$

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

$$\frac{4}{\frac{\frac{24}{155}}{4}}+\frac{y}{5}=42$$

Multiply both sides of the equation by 5.

$$\frac{775}{6}\times 4+y=210$$

Multiply \frac{775}{6} and 4 to get \frac{1550}{3}.

$$\frac{1550}{3}+y=210$$

Subtract \frac{1550}{3} from both sides.

$$y=210-\frac{1550}{3}$$

Subtract \frac{1550}{3} from 210 to get -\frac{920}{3}.

$$y=-\frac{920}{3}$$

The system is now solved.

$$x=\frac{24}{155} y=-\frac{920}{3}$$