Solve 2/x+5/y/frac{3{x}}=2/5

$$\frac { \frac { 2 } { x } + \frac { 5 } { y } } { \frac { 3 } { x } } = \frac { 2 } { 5 }$$

Solve for y

$$y=-\frac{25x}{4}$$

$$x\neq 0$$

Steps for Solving Linear Equation

Divide \frac{2}{x}+\frac{5}{y} by \frac{3}{x} by multiplying \frac{2}{x}+\frac{5}{y} by the reciprocal of \frac{3}{x}.

$$\frac{\left(\frac{2}{x}+\frac{5}{y}\right)x}{3}=\frac{2}{5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and y is xy. Multiply \frac{2}{x} times \frac{y}{y}. Multiply \frac{5}{y} times \frac{x}{x}.

$$\frac{\left(\frac{2y}{xy}+\frac{5x}{xy}\right)x}{3}=\frac{2}{5}$$

Since \frac{2y}{xy} and \frac{5x}{xy} have the same denominator, add them by adding their numerators.

$$\frac{\frac{2y+5x}{xy}x}{3}=\frac{2}{5}$$

Express \frac{2y+5x}{xy}x as a single fraction.

$$\frac{\frac{\left(2y+5x\right)x}{xy}}{3}=\frac{2}{5}$$

Cancel out x in both numerator and denominator.

$$\frac{\frac{5x+2y}{y}}{3}=\frac{2}{5}$$

Multiply both sides by 3.

$$\frac{5x+2y}{y}=\frac{2}{5}\times 3$$

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 5y, the least common multiple of y,5.

$$5\left(5x+2y\right)=\frac{2}{5}\times 3\times 5y$$

Cancel out 5 on both sides.

$$5x+2y=\frac{2}{5}\times 3y$$

Multiply \frac{2}{5} and 3 to get \frac{6}{5}.

$$5x+2y=\frac{6}{5}y$$

Subtract \frac{6}{5}y from both sides.

$$5x+2y-\frac{6}{5}y=0$$

Combine 2y and -\frac{6}{5}y to get \frac{4}{5}y.

$$5x+\frac{4}{5}y=0$$

Subtract 5x from both sides. Anything subtracted from zero gives its negation.

$$\frac{4}{5}y=-5x$$

Divide both sides of the equation by \frac{4}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{4}{5}y}{\frac{4}{5}}=-\frac{5x}{\frac{4}{5}}$$

Dividing by \frac{4}{5} undoes the multiplication by \frac{4}{5}.

$$y=-\frac{5x}{\frac{4}{5}}$$

Divide -5x by \frac{4}{5} by multiplying -5x by the reciprocal of \frac{4}{5}.

$$y=-\frac{25x}{4}$$

Variable y cannot be equal to 0.

$$y=-\frac{25x}{4}\text{, }y\neq 0$$

Solve for x

$$x=-\frac{4y}{25}$$

$$y\neq 0$$