Question

Simplify each expression

$3\left(5c+4d\right)+6\left(d-2c\right)$

Short answer

$3\left(5c+4d\right)+6\left(d-2c\right)$ is simplified to $3\left(c+6d\right)$

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 $a\ne 0$, 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

$\begin{array}{c}3\left(5c+4d\right)+6\left(d-2c\right)=15c+12d+6d-12c\\ =15c+12d-12c+6d\\ =15c-12c+12d+6d\\ =\left(15-12\right)c+\left(12+6\right)d\end{array}$

- Simplify parentheses

Simplify parentheses by adding the terms then use distributive property to take out common factor as shown below

$\begin{array}{c}\left(15-12\right)c+\left(12+6\right)d=3c+18d\\ =3\left(c+6d\right)\end{array}$