Question

Simplify each expression

51. $3\left(15x-9y\right)+5\left(4y-x\right)$

Short answer

$3\left(15x-9y\right)+5\left(4y-x\right)$ is simplified to $40x-7y$

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

$\begin{array}{c}3\left(15x-9y\right)+5\left(4y-x\right)=45x-27y+20y-5x\\ =45x-27y-5x+20y\\ =45x-5x-27y+20y\\ =\left(45-5\right)x+\left(-27+20\right)y\end{array}$

- Simplify further

To simplify, evaluate parentheses as shown below

$\left(45-5\right)x+\left(-27+20\right)y=40x-7y$