Chapter 1.4, Problem 61SR

To determine

To name: the property used in given equation.

Answer to Problem 61SR

The property used in given equation is Distributive property of Real numbers.

Explanation of Solution

Given information:

An equation is given as

  $z\left(9-4\right)=z\cdot 9-z\cdot 4$

Concept used:

For any real numbers $a,b$ , and $c$

real numbers properties are tabulated below:

Property	Addition	Multiplication
Commutative	$a+b=b+a$	$a\cdot b=b\cdot a$
Associative	$\left(a+b\right)+c=a+\left(b+c\right)$	$\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)$
Identity	$a+0=0+a=a$0 is identity element for addition	$a\cdot 1=1\cdot a=a$ 1 is identity element for Multiplication
Inverse	$a+\left(-a\right)=\left(-a\right)+a=0$$\left(-a\right)$ is inverse element for addition	$a\cdot \frac{1}{a}=\frac{1}{a}\cdot a=1$$\frac{1}{a}$ is inverse element for Multiplication
Closure	$a+b$ is real number	$a\cdot b$ is real number
Distributive	$\begin{array}{l}a\left(b+c\right)=ab+bc\\ \left(b+c\right)a=ba+ca\end{array}$	$\begin{array}{l}a\left(b+c\right)=ab+bc\\ \left(b+c\right)a=ba+ca\end{array}$

Calculation:

Consider the given equation.

  $z\left(9-4\right)=z\cdot 9-z\cdot 4$

Compare the above equation with $a\left(b-c\right)=a\cdot b-b\cdot c$ .

Here, Distributive property of Real numbers is used.