Chapter 4.7, Problem 27WE

To determine

To solve the equation $r=\frac{ab}{a+b}$ , for a .

Answer to Problem 27WE

  $a=\frac{rb}{b-r},b\ne r$

Explanation of Solution

Given:

The equation, $r=\frac{ab}{a+b}$

Concept Used:

• To get rid of a number in addition from one side, subtract the same number from both sides of equal sign.
• To get rid of a number in subtraction from one side, add the same number both sides of equal sign.
• To get rid of a number in multiplication from one side, divide the same number from both sides of equal sign.
• To get rid of a number in division from one side, multiply the same number both sides of equal sign.

Calculation:

To solve the given equation $r=\frac{ab}{a+b}$ for a , isolate the variable term a on one side by performing some basic algebraic operations to get rid of the other numbers and terms associated with it.

Here to isolate a on left side, first multiply both sides by $a+b$ and then simplify further by bringing the terms containing a on one side and solve further as shown below

  $\begin{array}{c}r=\frac{ab}{a+b}\\ r\cdot \left(a+b\right)=\frac{ab}{a+b}\cdot \left(a+b\right)\\ r\left(a+b\right)=ab\\ ra+rb=ab\\ ra+rb-ra=ab-ra\\ rb=a\left(b-r\right)\\ \Rightarrow a=\frac{rb}{b-r}\end{array}$

Thus, on solving for the indicated variable it gives $a=\frac{rb}{b-r}$ .

Since it is a rational expression after solving for the indicated variable, so the denominator should not be zero. That is, $b-r\ne 0\Rightarrow b\ne r$ .