Ontario 09 Academic (MPM1D)
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Add and subtract algebraic terms


Imagine this scenario

Zara drives to work at a shopping centre where she earns $\$12$$12 per hour, and pays $\$2$$2 per hour for parking.

The number of hours she works each week can vary. So if we let the variable $k$k represent how many hours she works in a week, we get:

  • Amount she earns each week = $12k$12k
  • Amount she pays in parking each week = $2k$2k

So each week, Zara earns $12k-2k$12k2k after paying for parking fees.


Can we simplify?

Can we express $12k-2k$12k2k more simply?

Since the value of both terms $12k$12k and $2k$2k depends on the same variable $k$k, they are like terms which we can combine.

We can think of $12k-2k$12k2k as subtracting $2$2 lots of $k$k from $12$12 lots of $k$k. This leaves us with $10$10 lots of $k$k.

That is, $12k-2k=10k$12k2k=10k

If we think about the scenario, Zara is really earning $\$10$$10 per hour that she works. For $k$k hours, she will earn $10k$10k.


Like terms

In mathematics, we want to express things in simplest form. In expressions where we are adding and subtracting terms, we can look to see if there are like terms we can combine.

When looking for like terms, the key is to look to see if the variable part of the terms is the same.

Example 1

 $5u$5u and $2u$2u are like terms. 

This is because $5u$5u is in terms of the variable $u$u, and $2u$2u is also in terms of the variable $u$u. So since they have the same variable (they both depend on $u$u), they are like terms.


$4t$4t and $8t^2$8t2 are NOT like terms because $4t$4t is in terms of variable $t$t, and $8t^2$8t2 is in terms of $t^2$t2.


As long as the different terms contain exactly the same variables, they are like terms, even if the order is different.

For example, $6mn$6mn and $9mn$9mn are like terms because the variables are the same.  $m\times n$m×n is exactly the same as $n\times m$n×m.

If we were to think of a like term for $4x^2y$4x2y that has a coefficient of $-2$2, we could say that either $-2x^2y$2x2y or $-2yx^2$2yx2 are like terms.


Adding and subtracting algebraic expressions

Now that we know what like terms are and that we can only add and subtract like terms, we can simplify algebraic expressions. It is important to remember that the operator directly before a number determines whether that number is positive or negative. You can look over how to add and subtract with positive and negative numbers if you need to.

Worked examples

question 1

Simplify $14x+8x$14x+8x 

Think: We can simplify this by adding the terms together because they are like terms. They both depend on $x$x.

Do: $14x+8x=22x$14x+8x=22x


question 2

Question: What algebraic expression should go in the space to make this statement true?

$15p$15p - ? = $5p$5p

Think: $15p$15p and $5p$5p are like terms, so how many lots of $p$p would we need to take away from $15p$15p to get back to $5p$5p?

Do: $15p-10p=5p$15p10p=5p


question 3

Question: Simplify the expression: $10a-8b-4a-2b$10a8b4a2b

Think: Let's group the like terms: 10a - 8b - 4a - 2b 

Combining the like terms we get:

$10a-4a$10a4a = $6a$6a

$-8b-2b$8b2b = $-10b$10b

Do: $10a-8b-4a-2b=6a-10b$10a8b4a2b=6a10b 


 Question 4

Are the following like terms? $4xy$4xy and $10yx$10yx

  1. Yes








Question 5

Simplify the expression:


Question 6

Simplify the expression:




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