Question

Simplify each expression

$\frac{1}{6}\left(3x+5y\right)+\frac{2}{3}\left(\frac{3}{5}x-6y\right)$

Short answer

$\frac{1}{6}\left(3x+5y\right)+\frac{2}{3}\left(\frac{3}{5}x-6y\right)$ is simplified to $\frac{9}{10}x-\frac{19}{6}y$

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

$\begin{array}{c}\frac{1}{6}\left(3x+5y\right)+\frac{2}{3}\left(\frac{3}{5}x-6y\right)=\frac{3}{6}x+\frac{5}{6}y+\frac{2}{5}x-4y\\ =\frac{3}{6}x+\frac{2}{5}x+\frac{5}{6}y-4y\\ =\left(\frac{3}{6}+\frac{2}{5}\right)x+\left(\frac{5}{6}-4\right)y\end{array}$

- Simplify further 

To simplify, evaluate parentheses as shown below

$\begin{array}{c}\left(\frac{3}{6}+\frac{2}{5}\right)x+\left(\frac{5}{6}-4\right)y=\left(\frac{\left(3\cdot 5\right)+\left(2\cdot 6\right)}{30}\right)x+\left(\frac{5-\left(4\cdot 6\right)}{6}\right)y\\ =\left(\frac{15+12}{30}\right)x+\left(\frac{5-24}{6}\right)y\\ =\left(\frac{27}{30}\right)x+\left(\frac{-19}{6}\right)y\\ =\frac{9}{10}x-\frac{19}{6}y\end{array}$