Chapter 7.2, Problem 40IP

To determine

To simplify the expression $\frac{2}{3}\left(6a+3b\right)-\frac{1}{2}\left(a-2b\right)$

Answer to Problem 40IP

  $\frac{21a+18b}{6}$

Explanation of Solution

Given information:

The expression, $\frac{2}{3}\left(6a+3b\right)-\frac{1}{2}\left(a-2b\right)$

Concept used:

• The distributive property of binomials (FOIL): $\left(a+b\right)\left(c+d\right)=a\cdot c+a\cdot d+b\cdot c+b\cdot d$
• Left Distributive property: $a\left(b+c\right)=a\cdot b+a\cdot c$
• Exponent property: $x^n\cdot x^m=x^{n+m}$

Calculation:

In order to simplify the given expression, use the left distributive property and simplify the like terms as shown below

  $\begin{array}{c}\frac{2}{3}\left(6a+3b\right)-\frac{1}{2}\left(a-2b\right)\\ =\frac{2}{3}\cdot 6a+\frac{2}{3}\cdot 3b-\left(\frac{1}{2}\cdot a+\frac{1}{2}\cdot \left(-2b\right)\right)\\ =\frac{12a}{3}+\frac{6b}{3}-\frac{a}{2}+\frac{2b}{2}\\ =\frac{12a}{3}\cdot \frac{2}{2}+\frac{6b}{3}\cdot \frac{2}{2}-\frac{a}{2}\cdot \frac{3}{3}+\frac{2b}{2}\cdot \frac{3}{3}\\ =\frac{24a}{6}+\frac{12b}{6}-\frac{3a}{6}+\frac{6b}{6}\\ =\frac{24a+12b-3a+6b}{6}\\ =\frac{21a+18b}{6}\end{array}$

Thus, the given expression simplifies to $\frac{21a+18b}{6}$