Q33. - Algebra 2

Question

Name the property illustrated by each equation. 

$\left(1\frac{2}{7}\right)\left(\frac{7}{9}\right)=1$

Step-by-Step Solution

Answer

Name of the property illustrated is Multiplicative Inverse.

Step 1 - Write properties of real numbers

For any real numbers $a,b,$ and $c:$
Property	Addition	Multiplication
Commutative	$a+b=b+a$	$a·b=b·a$
Associative	$\left(a+b\right)+c=a+\left(b+c\right)$	$\left(a·b\right)·c=a·\left(b·c\right)$
Identity	$a+0=a=0+a$	$a·1=a=1·a$
Inverse	$a+\left(-a\right)=0=\left(-a\right)+a$	If $a\ne 0$, then $a·\frac{1}{a}=1=\frac{1}{a}·a$
Distributive	$a\left(b+c\right)=ab+ac\text{ and }\left(b+c\right)a=ba+ca$

Step 2 - Substitution in given values

First simplify mixed fraction

$\begin{array}{c}1\frac{2}{7}=1+\frac{2}{7}\\ =\frac{7}{7}+\frac{2}{7}\\ =\frac{7+2}{7}\\ =\frac{9}{7}\end{array}$

Substitute a for $\frac{9}{7}$

As $\frac{7}{9}$ can be written as $\frac{1}{\frac{9}{7}}$ which further can be written as $\frac{1}{a}$

Now given equation becomes $a·\frac{1}{a}=1$

Step 3 - Identify the property

From the table of properties of real numbers, $a·\frac{1}{a}=1$ is the property of inverse in multiplication

Thus, multiplicative inverse is illustrated by given equation.