Solve x/5-x=4/5x/4-frac{4{5}x}

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

Solve for x

$$x\neq 5$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{\frac{4}{5}x}{4-\frac{4}{5}x}.

$$\frac{x}{5-x}=\frac{\frac{4}{5}x}{\frac{4}{5}\left(-x+5\right)}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{x}{5-x}=\frac{x}{\left(\frac{4}{5}\right)^{0}\left(-x+5\right)}$$

Calculate \frac{4}{5} to the power of 0 and get 1.

$$\frac{x}{5-x}=\frac{x}{1\left(-x+5\right)}$$

Use the distributive property to multiply 1 by -x+5.

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

Subtract \frac{x}{-x+5} from both sides.

$$\frac{x}{5-x}-\frac{x}{-x+5}=0$$

Since \frac{x}{5-x} and \frac{x}{-x+5} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x-x}{5-x}=0$$

Combine like terms in x-x.

$$\frac{0}{5-x}=0$$

Variable x cannot be equal to 5 since division by zero is not defined. Multiply both sides of the equation by -x+5.

$$0=0$$

This is true for any x.

$$x\in \mathrm{R}$$

Variable x cannot be equal to 5.

$$x\in \mathrm{R}\setminus 5$$