After having studied what algebraic expressions and equivalent algebraic expressions are, the next thing to do is to understand how to collect like terms.
Since the numbers and variables are not similar (or, 'like') terms, they cannot be simplified into a single group and, therefore, we have to write them separately (X,Y).
Example with Two Variables
The expression 4+2+2X+3X+Y+2Y= can be simplified as follows:
5X+3Y+6
Practice Exercises: Collecting Like Terms
Combine like terms in order to obtain shorter expressions:
- X+X=
- 5+8β9+5Xβ4X=
- 5+0+8Xβ5=
- 11+5Xβ2X+8=
- 13X+5β4.5X+7.5X=
Collect like terms to obtain shorter expressions. Then, using what we have learned about the numerical value of algebraic expressions, apply X=5 and solve.
- 2X+5Xβ
4=
- 2.3X+0.4Xβ0.7X=
- 15Xβ+15Xβ=
- 83βXβ82βX+5=
- (7+Y):3=
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Exercises: Collecting Like Terms
Exercise 1
Task:
3abββ
183βa+85βb+184βm+109βa+32βm=?
Solution:
Enter the corresponding elements.
3abβΓ183βa+85βb+109βa+184βm+32βm=
Convert the mixed fractions into improper fractions.
3abβΓ8(8+3)βa+85βb+109βa+184βm+32βm=
Solve accordingly.
8Γa3Γ11ΓbΓaβ+85βb+109βa+184+2Γ6βm=
Simplify to a in the equation.
833βb+85βb+109βa+1816βm=
833+5βb+109βa+98βm=
838βb+109βa+98βm=
443βb+109βa+98βm=
Answer:
443βb+109βa+98βm=
Exercise 2
Task:
83βa+914βb+191βb+86βa=?
Solution:
First, group the terms together:
83βa+86βa+914βb+191βb
Then reduce in correspondence and convert the mixed fractions into improper fractions.
83+6βa+910βb+914βb=
89βa+910+14βb=
181βa+924βb=
181βa+296βb=
181βa+232βb
Answer:
181βa+232βb
Do you know what the answer is?
Exercise 3
7.3Γ4a+2.3+8a=?
Solution:
We start with the multiplication operation.
(7.3Γ4a)+2.3+8a=
(29.2a)+2.3+8a=
Then we add together as much as we can and rewrite the equation to make it clearer:
29.2a+8a+2.3=
37.2a+2.3=
Answer:
37.2a+2.3
Exercise 4
Task:
Solve the following equation:
a+b+bc+9a+10b+3c=?
Solution:
The terms are substituted into the expression according to the order: a,b,c.
a+9a+b+bc+10b+3c=
We continue with the addition operations.
10a+11b+bc+3c=
The terms containing c are converted for the equation since they cannot be simplified further.
10a+11b+(b+3)c
Answer:
10a+11b+(b+3)c
Exercise 5
Task:
Solve the following equation:
3z+19zβ4z=?
Solution:
We start with the addition operation:
22zβ4z=
We continue solving accordingly.
18z
Answer:
18z
Review Questions
What are like terms?
Like terms in an algebraic expression are those that have the same variable with the same exponent, regardless of the sign and the coefficientβthat is, the sign and the coefficient can be different, but the variable and the exponent must be the same. For example:
3x2 y β11x2
8a5 y 8a5
β7m y 32βm
How do you simplify expressions?
In order to simplify algebraic expressions, we need to work out if there are any like terms and then group them together before performing the operations (addition, subtraction, etc.).
Example 1
Task: Simplify the following expression:
3x2β7x+6β5x2βxβ1
Solution:
First we need to group the like terms and then perform the operations.
3x2β5x2β7xβx+6β1
β2x2β8x+5
Answer:
β2x2β8x+5
Example 2
Task: Simplify the following expression:
8m2+2m+7=3m3+5m2+2mβ5
Solution:
In this case we need to put all the terms on one side and combine the like terms:
β3m3+8m2β5m2+2mβ2m+7+5=
β3m3+3m2+12=
Answer:
β3m3+3m2+12=
How do you simplify a function?
In order to simplify a function, we must also work out if there are any like terms and group them together in order to simplify them.
Example
Question: Simplify the following function:
f(a)=β5a2+2aβ4+2a2+5a
Solution:
In this function we can see that there are like terms, so we can group them to simplify them.
f(a)=β5a2+2a2+2a+5aβ4
f(a)=β3a2+7aβ4
Answer:
f(a)=β3a2+7aβ4
Do you think you will be able to solve it?
Examples with solutions for Simplifying Expressions (Collecting Like Terms)
Exercise #1
18xβ7+4xβ9β8x=?
Video Solution
Step-by-Step Solution
To solve the exercise, we will reorder the numbers using the substitution property.
18xβ8x+4xβ7β9=
To continue, let's remember an important rule:
1. It is impossible to add or subtract numbers with variables.
That is, we cannot subtract 7 from 8X, for example...
We solve according to the order of arithmetic operations, from left to right:
18xβ8x=10x10x+4x=14xβ7β9=β16Remember, these two numbers cannot be added or subtracted, so the result is:
14xβ16
Answer
Exercise #2
7.3β
4a+2.3+8a=?
Video Solution
Step-by-Step Solution
It is important to remember that when we have numbers and variables, it is impossible to add or subtract them from each other.
We group the elements:
7.3Γ4a+2.3+8a=
29.2a + 2.3 + 8a =
37.2a+2.3
And in this exercise, this is the solution!
You can continue looking for the value of a.
But in this case, there is no need.
Answer
37.2a+2.3
Exercise #3
3m29mβΓ63mβ=
Video Solution
Step-by-Step Solution
According to the laws of multiplication, we must first simplify everything into one exercise:
3m2Γ69mΓ3mβ=
We will simplify and get:
m2Γ69m2β=
We will simplify and get:
69β=
We will factor the expression into a multiplication:
3Γ23Γ3β=
We will simplify and get:
23β=1.5
Answer
Exercise #4
Are the expressions the same or not?
18x
2+9x
Video Solution
Answer
Exercise #5
Are the expressions the same or not?
20x
2Γ10x
Video Solution
Answer