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# 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 + 3y has two terms.

First term: 5x, has factors 5 and x

Second term: 3y, has factors 3 and y

The 5 and 3 are called the coefficients of the terms.

### Example 3

The expression

3x^2 - 7ab + 2esqrt(pi)

has three terms.

First term: 3x^2 has factors 3 and x2

Second term: -7ab has factors -7, a and b

Third Term: 2esqrt(pi); has factors 2, e, and sqrt(pi).

The 3, -7 and 2 are called coefficients of the terms.

## 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 13x and -2x. We cannot do anything with the 7y or 6a.

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.

### Example 7

Simplify −5[−2(m − 3n) + 4n]

Go back to Order of Operations if you are not sure what to do first with this question.

−5[−2(m − 3n) + 4n]

The square brackets [ ] work just the same as round brackets ( ). We could have used curly brackets { } here as well.

The first thing we do is expand out the round brackets inside.

−2(m − 3n) = −2m + 6n

The negative times negative in the middle gives positive 6n.

Now add the 4n in the square brackets:

[−2m + 6n + 4n] = [−2m + 10n]

Remembering the −5 out front, our problem has become:

−5[−2m + 10n] = 10m − 50n

Taking each term one at a time, what we did was:

−5 × −2m = 10m (Two negative numbers multiplied together give a positive); and

−5 × 10n = −50n (Negative times positive gives negative)

Go back to the section on Integers if you are not sure about multiplying with negative numbers.

So here's the answer:

−5[−2(m − 3n) + 4n] = 10m − 50n

Note:

The fancy name for round brackets ( ) is "parentheses".

The fancy name for square brackets [ ] is "box brackets".

The fancy name for curly brackets { } is "braces".

### Example 8

Simplify −[7(a − 2b) − 4b]

−[7(a − 2b) − 4b]

= −[7a − 14b − 4b]

= −[7a − 18b]

= −7a + 18b