Q8. - Algebra 2

Question

8. Name the property illustrated by $bc+\left(-bc\right)=0$

Step-by-Step Solution

Answer

Name of the property illustrated is Additive Inverse.

Step 1 - Write properties of real numbers

Properties of real numbers are written in following table:

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

Substitute a for bc in given equation.

For a=bc

Multiply both sides by $-1$

$\begin{array}{c}\left(-1\right)\times a=\left(-1\right)\times bc\\ -a=-bc\end{array}$

Now equation becomes $a+\left(-a\right)=0$

Step 3 - Identify the property

From the table of properties of real numbers, $a+\left(-a\right)=0$ is the property of inverse in addition

Thus, additive inverse is illustrated by given equation.