Question

Simplify each expression

$\frac{1}{2}\left(16-4a\right)-\frac{3}{4}\left(12+20a\right)$

Short answer

$\frac{1}{2}\left(16-4a\right)-\frac{3}{4}\left(12+20a\right)$ is simplified to $-1-17a$

Step-by-step solution

- 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	$a\left(b+c\right)=ab+ac\text{and}\left(b+c\right)a=ba+ca$

- Simplify by grouping same variables

To simplify given equation, use commutative and distributive property to group coefficients of same variable as shown below

$\begin{array}{c}\frac{1}{2}\left(16-4a\right)-\frac{3}{4}\left(12+20a\right)=\left(\frac{1}{2}\cdot 16\right)-\left(\frac{1}{2}\cdot 4a\right)-\left(\frac{3}{4}\cdot 12\right)-\left(\frac{3}{4}\cdot 20a\right)\\ =8-2a-9-15a\\ =8-9-2a-15a\\ =\left(8-9\right)-\left(2+15\right)a\end{array}$

- Simplify parentheses

Simplify parentheses by adding or subtracting the terms as shown below

$\left(8-9\right)-\left(2+15\right)a=-1-17a$