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
Answer
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]
Answer
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]
Answer
−[7(a − 2b) − 4b]
= −[7a − 14b − 4b]
= −[7a − 18b]
= −7a + 18b