Simplifying Expressions

Collecting Like and Unlike Terms:

Like terms are terms whose variables (and their exponents such as the 2 in a²) are the same.

In other words, terms that are “like” each other. If they are not like terms, they are called, Unlike Terms.

Note: the coefficients (the numbers you multiply by, such as “5” in 5x) can be different.

Consider terms as 8y, 9y, 5y, and -2y

Here all four terms are like terms because y is the common variable.

Consider another example; \( \scriptsize 3xy^2,\: \normalsize \frac{1}{2} \scriptsize xy^2,\: 7xy^2 \: and \: \normalsize \frac{3}{4} \scriptsize xy^2 \)

Here also all four terms are like terms because xy²  is the common variable.

In the above example, 5xy is not a like term because xy is not raised to the power of 2.

Example 2.2.1:

Simplify:

(a) \( \scriptsize 4a \: + \: 3b \: + \: 2a \: + \: b \)
(b) \( \scriptsize 8x \: + \: 5y \: – \: 5x \: – \: 3y\: + \: 1 \)

Solution

(a) \( \scriptsize 4a \: + \: 3b \: + \: 2a \: + \: b \)

Collecting like “a” terms together and then the “b” terms to get

\( \scriptsize 4a \: + \: 2a \: + \: 3b \: + \: b \)

6a   +   4b

(b) \( \scriptsize 8x \: + \: 5y \: – \: 5x \: – \: 3y\: + \: 1 \)

Collecting the like term by arranging the terms by their variables.

\( \scriptsize 8x \: – \: 5x \: + \: 5y \: – \: 3y\: + \: 1 \)

3x   +  2y  +  1

Example 2.2.2:

Simplify

(a) \( \scriptsize \: -5x \: + \: 9x \: + \: 2 \: – \: x \)
(b) \( \scriptsize \: -3xz \: + \: 7xy \: + \: 2xa \: – \: 9xz\: + \: 4xa \)

Solution

(a) \( \scriptsize \: -5x \: + \: 9x \: + \: 2 \: – \: x \)

Collect like terms

\( \scriptsize \: 9x \: – \: 5x \: – \: x \: + \: 2 \)

\( \scriptsize \: 4x \: – \: x \: + \: 2 \)

\( \scriptsize = 3x \: + \: 2 \)

(b) \( \scriptsize \: -3xz \: + \: 7xy \: + \: 2xa \: – \: 9xz\: + \: 4xa \)

Collect like terms

\( \scriptsize \: -9xz \: – \: 3xz \: + \: 2xa \: + \: 4xa\: + \: 7xy\)

= \( \scriptsize \: -12xz \: + \: 6xa \: + \: 7xy\)

or \( \scriptsize \: 6xa\: – \: 12xz \: + \: 7xy\)

Multiplying with Letters:

Note that, the multiplication of numbers is regarded as repeated addition. 

e.g.   4  +  4  +  4   =  12

Note that 4 is repeated 3 times so we can also say

4  +  4  +  4  =   4  ×  3   =  12

Similarly, in algebraAlgebra is a branch of mathematics that substitutes letters for numbers. Algebra is about finding the unknown or putting real-life variables into equations and then solving them. More, we can also say

\( \scriptsize x \: + \: x\: + \: x\: + \: x\: + \: x = 5x \)

Also,

\( \scriptsize 5 \: \times \: x = 5x \)

\( \scriptsize 7 \: \times \: p = 7p \)

\( \scriptsize a \: \times \: b = ab \)

\( \scriptsize x \: \times \: y \: \times \: 3 = 3xy \)

Also,

\( \scriptsize 25ab = 25 \: \times \: a \: \times \: b \)

\( \scriptsize 2xyz = 2 \: \times \: x \: \times \: y \: \times \: z \)