Solve 1/v-1/u=m-1/r

$$\frac { 1 } { v } - \frac { 1 } { u } = \frac { m - 1 } { r }$$

Solve for m

$$m=\frac{ru-rv+uv}{uv}$$

$$v\neq 0\text{ and }u\neq 0\text{ and }r\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by ruv, the least common multiple of v,u,r.

$$ru-rv=uv\left(m-1\right)$$

Use the distributive property to multiply uv by m-1.

$$ru-rv=uvm-uv$$

Swap sides so that all variable terms are on the left hand side.

$$uvm-uv=ru-rv$$

Add uv to both sides.

$$uvm=ru-rv+uv$$

Divide both sides by uv.

$$\frac{uvm}{uv}=\frac{ru-rv+uv}{uv}$$

Dividing by uv undoes the multiplication by uv.

$$m=\frac{ru-rv+uv}{uv}$$

Divide ru-rv+uv by uv.

$$m=\frac{r}{v}-\frac{r}{u}+1$$

Solve for r

$$\left\{\begin{matrix}r=-\frac{uv\left(1-m\right)}{u-v}\text{, }&m\neq 1\text{ and }u\neq 0\text{ and }v\neq 0\text{ and }v\neq u\\r\neq 0\text{, }&u=v\text{ and }m=1\text{ and }v\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ru-rv=uv\left(m-1\right)$$

Use the distributive property to multiply uv by m-1.

$$ru-rv=uvm-uv$$

Reorder the terms.

$$ru-rv=muv-uv$$

Combine all terms containing r.

$$\left(u-v\right)r=muv-uv$$

Divide both sides by u-v.

$$\frac{\left(u-v\right)r}{u-v}=\frac{uv\left(m-1\right)}{u-v}$$

Dividing by u-v undoes the multiplication by u-v.

$$r=\frac{uv\left(m-1\right)}{u-v}$$

Variable r cannot be equal to 0.

$$r=\frac{uv\left(m-1\right)}{u-v}\text{, }r\neq 0$$