Chapter 5.7, Problem 42E

To determine

Tofind:The solution of equation $\frac{2}{3}x-1=\frac{5}{6}$ by using multiplicative inverse and method of clearing fractions and compare.

Answer to Problem 42E

The solution of equation $\frac{2}{3}x-1=\frac{5}{6}$ is $x=\frac{11}{4}$ .

Explanation of Solution

Given information: Consider, the equation $\frac{2}{3}x-1=\frac{5}{6}$ provided in the question.

Calculation:

Multiplicative inverse method:

To evaluate the provided equation $\frac{2}{3}x-1=\frac{5}{6}$ , it is needed to add $1$ in both side of the equation

Therefore,

  $\frac{2}{3}x-1=\frac{5}{6}$

  $\begin{array}{l}\Rightarrow \frac{2}{3}x-1+1=\frac{5}{6}+1\\ \Rightarrow \frac{2}{3}x=\frac{5}{6}+1\\ \Rightarrow \frac{2}{3}x=\frac{5+6}{6}\\ \Rightarrow \frac{2}{3}x=\frac{11}{6}\end{array}$

  $\text{Multiply}\text{each}\text{side}\text{by}\text{multiplicative}\text{inverse}\text{of}\frac{\text{2}}{\text{3}}$

  $\begin{array}{l}\Rightarrow \frac{2x}{3}\times \frac{3}{2}=\frac{11}{6}\times \frac{3}{2}\\ \Rightarrow x=\frac{11}{4}\end{array}$

Clearing fractions method:

To evaluate the provided equation $\frac{2}{3}x-1=\frac{5}{6}$ , it is needed to multiply both side by least common denominator $6$ .

Therefore,

  $\Rightarrow 6(\frac{2}{3}x-1)=\frac{5}{6}\times 6$

  $\text{Using}\text{distributive}\text{property}$ ,

  $\begin{array}{l}\Rightarrow 6\times \frac{2}{3}x-1\times 6=5\\ \Rightarrow 4x-6=5\end{array}$

  $\text{Add}\text{6}\text{in}\text{each}\text{side}$

  $\begin{array}{l}\Rightarrow 4x-6+6=5+6\\ \Rightarrow 4x=11\end{array}$

  $\text{Divide}\text{each}\text{side}\text{by}\text{4}$

  $\begin{array}{l}\Rightarrow \frac{4x}{4}=\frac{11}{4}\\ \Rightarrow x=\frac{11}{4}\end{array}$

So, the value of $x$ is same by both method.

Hence,

The solution of equation $\frac{2}{3}x-1=\frac{5}{6}$ is $x=\frac{11}{4}$