Question

Simplify each expression

53. $8\left(r+7t\right)-4\left(13t+5r\right)$

Short answer

$8\left(r+7t\right)-4\left(13t+5r\right)$ is simplified to $-4\left(3r-t\right)$

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$$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, first use distributive property then commutative property of addition then distributive property to group coefficients of same variable as shown below

$\begin{array}{c}8\left(r+7t\right)-4\left(13t+5r\right)=8r+56t-52t-20r\\ =8r+56t-20r-52t\\ =8r-20r+56t-52t\\ =\left(8-20\right)r+\left(56-52\right)t\end{array}$

- Use distributive property 

Firstly simplify parentheses then take out common factor of both terms as shown below

$\begin{array}{c}\left(8-20\right)r+\left(56-52\right)t=-12r+4t\\ =-4\left(3r-t\right)\end{array}$