algebric addition and subtraction

1. Addition and Subtraction of Algebraic Expressions

Before we see how to add and subtract integers, we define terms and factors.

Terms and Factors

A term in an algebraic expression is an expression involving letters and/or numbers (called factors), multiplied together.

Example 1

The algebraic expression

5x

is an example of one single term. It has factors 5 and x.

The 5 is called the coefficient of the term and the x is a variable.

Example 2

5x + 3yhas two terms.

First term:5x, has factors $\displaystyle{5}$ and x
Second term: 3y, has factors $\displaystyle{3}$ and y

The $\displaystyle{5}$ and $\displaystyle{3}$ are called the coefficients of the terms.

Example 3

The expression

$\displaystyle{3}{x}^{2}-{7}{a}{b}+{2}{e}\sqrt{{\pi}}$

has three terms.

First term: $\displaystyle{3}{x}^{2}$ has factors $\displaystyle{3}$ and x²
Second term: $\displaystyle-{7}{a}{b}$ has factors $\displaystyle-{7}$, a and b
Third Term: $\displaystyle{2}{e}\sqrt{{\pi}}$; has factors $\displaystyle{2}$, $\displaystyle{e}$, and $\displaystyle\sqrt{{\pi}}$.

The $\displaystyle{3}$, $\displaystyle-{7}$ and $\displaystyle{2}$ are called coefficients of the terms.

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Like Terms

"Like terms" are terms that contain the same variables raised to the same power.

Example 4

3x² and 7x² are like terms.

Example 5

-8x² and 5y² are not like terms, because the variable is not the same.

Adding and Subtracting Terms

Important: We can only add or subtract like terms.

Why? Think of it like this. On a table we have 4 pencils and 2 books. We cannot add the 4 pencils to the 2 books - they are not the same kind of object.

We go get another 3 pencils and 6 books. Altogether we now have 7 pencils and 8 books. We can't combine these quantities, since they are different types of objects.

Next, our sister comes in and grabs 5 pencils. We are left with 2 pencils and we still have the 8 books.

Similarly with algebra, we can only add (or subtract) similar "objects", or those with the same letter raised to the same power.

Example 6

Simplify 13x + 7y − 2x + 6a

13x + 7y − 2x + 6a
The only like terms in this expression are $\displaystyle{13}{x}$and $\displaystyle-{2}{x}$. We cannot do anything with the $\displaystyle{7}{y}$or $\displaystyle{6}{a}$.
So we group together the terms we can subtract, and just leave the rest:

(13x − 2x) + 6a + 7y
= 6a + 11x + 7y

We usually present our variables in alphabetical order, but it is not essential.