Simplifying Algebraic Expressions

Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.

Example 12: Simplifying Algebraic Expressions

Simplify each algebraic expression.

$3x - 2y+x - 3y - 7$
$2r - 5\left(3-r\right)+4$
$\left(4t-\frac{5}{4}s\right)-\left(\frac{2}{3}t+2s\right)$
$2mn - 5m+3mn+n$

Solution

$\begin{array}\text{ }3x-2y+x-3y-7 \hfill& =3x+x-2y-3y-7 \hfill& \text{Commutative property of addition} \\ \hfill& =4x-5y-7 \hfill& \text{Simplify}\end{array}$
$\begin{array}2r-5\left(3-r\right)+4 \hfill& =2r-15+5r+4 \hfill& \text{Distributive property} \\ \hfill& =2r+5y-15+4 \hfill& \text{Commutative property of addition} \\ \hfill& =7r-11 \hfill& \text{Simplify}\end{array}$
$\begin{array}4t-4\left(t-\frac{5}{4}s\right)-\left(\frac{2}{3}t+2s\right) \hfill& =4t-\frac{5}{4}s-\frac{2}{3}t-2s \hfill& \text{Distributive property} \\ \hfill& =4t-\frac{2}{3}t-\frac{5}{4}s-2s \hfill& \text{Commutative property of addition} \\ \hfill& =\text{10}{3}t-\frac{13}{4}s \hfill& \text{Simplify}\end{array}$
$\begin{array}\text{ }mn-5m+3mn+n \hfill& =2mn+3mn-5m+n \hfill& \text{Commutative property of addition} \\ \hfill& =5mn-5m+n \hfill& \text{Simplify}\end{array}$

Try It 12

Simplify each algebraic expression.

$\frac{2}{3}y - 2\left(\frac{4}{3}y+z\right)$
$\frac{5}{t}-2-\frac{3}{t}+1$
$4p\left(q - 1\right)+q\left(1-p\right)$
$9r-\left(s+2r\right)+\left(6-s\right)$

Solution

Example 13: Simplifying a Formula

A rectangle with length $L$ and width $W$ has a perimeter $P$ given by $P=L+W+L+W$ . Simplify this expression.

Solution

\begin{matrix} P = L + W + L + W \\ P = L + L + W + W & Commutative property of addition \\ P = 2 L + 2 W & Simplify \\ P = 2 (L + W) & Distributive property \end{matrix}

$\begin{array}\text{ }P=L+W+L+W \\ P=L+L+W+W \hfill& \text{Commutative property of addition} \\ P=2L+2W \hfill& \text{Simplify} \\ P=2\left(L+W\right) \hfill& \text{Distributive property}\end{array}$

Try It 13

If the amount $P$ is deposited into an account paying simple interest $r$ for time $t$ , the total value of the deposit $A$ is given by $A=P+Prt$ . Simplify the expression. (This formula will be explored in more detail later in the course.)

Solution

Real Numbers: Algebra Essentials