Simplifying Expressions (Collecting Like Terms)

The simplification of expressions consists of creating an equivalent expression written in a shorter and simpler way in which we combine all of the similar terms (collecting like terms).

For example, the expression:

$3+3+3+3+3+5X-3X$

After having simplified it, it would be:

$15+2X$

What we have done is created two groups of numbers and variables:
$3+3+3+3+3$ and $5X-3X$ .

This can be simplified further, resulting in only two terms: $15+2X$

Detailed explanation

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Test yourself on variables and algebraic expressions!

$3x+4x+7+2=\text{?}$

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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:

$X+X=$
$5+8-9+5X-4X=$
$5+0+8X-5=$
$11+5X-2X+8=$
$13X+5-4.5X+7.5X=$

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.

$2X+5X\cdot4=$
$2.3X+0.4X-0.7X=$
${X\over15}+{X\over15}=$
${3\over8}X-{2\over8}X+5=$
$(7+Y):3=$

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Test your knowledge

Question 1

$3z+19z-4z=\text{?}$

Question 2

$$ 11+5x-2x+8= $$

Question 3

$$ 5+0+8x-5= $$

Exercises: Collecting Like Terms

Exercise 1

Task:

$3\frac{b}{a}\cdot1\frac{3}{8}a+\frac{5}{8}b+\frac{4}{18}m+\frac{9}{10}a+\frac{2}{3}m=\text{?}$

Solution:

Enter the corresponding elements.

$3\frac{b}{a}\times1\frac{3}{8}a+\frac{5}{8}b+\frac{9}{10}a+\frac{4}{18}m+\frac{2}{3}m=$

Convert the mixed fractions into improper fractions.

$3\frac{b}{a}\times\frac{(8+3)}{8}a+\frac{5}{8}b+\frac{9}{10}a+\frac{4}{18}m+\frac{2}{3}m=$

Solve accordingly.

$\frac{3\times11\times b\times a}{8\times a}+\frac{5}{8}b+\frac{9}{10}a+\frac{4+2\times6}{18}m=$

Simplify to $a$ in the equation.

$\frac{33}{8}b+\frac{5}{8}b+\frac{9}{10}a+\frac{16}{18}m=$

$\frac{33+5}{8}b+\frac{9}{10}a+\frac{8}{9}m=$

$\frac{38}{8}b+\frac{9}{10}a+\frac{8}{9}m=$

$4\frac{3}{4}b+\frac{9}{10}a+\frac{8}{9}m=$

Answer:

$4\frac{3}{4}b+\frac{9}{10}a+\frac{8}{9}m=$

Exercise 2

Task:

$\frac{3}{8}a+\frac{14}{9}b+1\frac{1}{9}b+\frac{6}{8}a=\text{?}$

Solution:

First, group the terms together:

$\frac{3}{8}a+\frac{6}{8}a+\frac{14}{9}b+1\frac{1}{9}b$

Then reduce in correspondence and convert the mixed fractions into improper fractions.

$\frac{3+6}{8}a+\frac{10}{9}b+\frac{14}{9}b=$

$\frac{9}{8}a+\frac{10+14}{9}b=$

$1\frac{1}{8}a+\frac{24}{9}b=$

$1\frac{1}{8}a+2\frac{6}{9}b=$

$1\frac{1}{8}a+2\frac{2}{3}b$

Answer:

$1\frac{1}{8}a+2\frac{2}{3}b$

Do you know what the answer is?

Question 1

$$ 5+8-9+5x-4x= $$

Question 2

$$ x+x= $$

Question 3

Are the expressions the same or not?

$$ 18x $$

$$ 2+9x $$

Exercise 3

$7.3\times4a+2.3+8a=?$

Solution:

We start with the multiplication operation.

$(7.3\times4a)+2.3+8a=$

$(29.2a)+2.3+8a=$

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:

$a+b+bc+9a+10b+3c=\text{?}$

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:

$10a+11b+(b+3)c$

Check your understanding

Question 1

Are the expressions the same or not?

$$ 3+3+3+3 $$

$3\times4$

Question 2

Are the expressions the same or not?

$$ 20x $$

$2\times10x$

Question 3

$7a+8b+4a+9b=\text{?}$

Exercise 5

Task:

Solve the following equation:

$3z+19z-4z=\text{?}$

Solution:

We start with the addition operation:

$22z-4z=$

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 coefficient—that is, the sign and the coefficient can be different, but the variable and the exponent must be the same. For example:

$3x^2$ y $-11x^2$

$8a^5$ y $8a^5$

$-7m$ y $\frac{2}{3}m$

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:

$3x^2-7x+6-5x^2-x-1$

Solution:

First we need to group the like terms and then perform the operations.

$3x^2-5x^2-7x-x+6-1$

$-2x^2-8x+5$

Answer:

$-2x^2-8x+5$

Example 2

Task: Simplify the following expression:

$8m^2+2m+7=3m^3+5m^2+2m-5$

Solution:

In this case we need to put all the terms on one side and combine the like terms:

$-3m^3+8m^2-5m^2+2m-2m+7+5=$

$-3m^3+3m^2+12=$

Answer:

$-3m^3+3m^2+12=$

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:

$f\left(a\right)=-5a^2+2a-4+2a^2+5a$

Solution:

In this function we can see that there are like terms, so we can group them to simplify them.

$f\left(a\right)=-5a^2+2a^2+2a+5a-4$

$f\left(a\right)=-3a^2+7a-4$

Answer:

$f\left(a\right)=-3a^2+7a-4$

Do you think you will be able to solve it?

Question 1

$18x-7+4x-9-8x=\text{?}$

Question 2

^{$13a+14b+17c-4a-2b-4b=\text{?}$}

Question 3

^{$a+b+bc+9a+10b+3c=\text{?}$}

Examples with solutions for Simplifying Expressions (Collecting Like Terms)

Exercise #1

$18x-7+4x-9-8x=\text{?}$

Video Solution

Step-by-Step Solution

To solve the exercise, we will reorder the numbers using the substitution property.

$18x-8x+4x-7-9=$

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:

$18x-8x=10x$ $10x+4x=14x$ $-7-9=-16$ Remember, these two numbers cannot be added or subtracted, so the result is:

$14x-16$

Answer

$14x-16$

Exercise #2

$7.3\cdot4a+2.3+8a=\text{?}$

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:

$7.3×4a + 2.3 + 8a =$

$29.2a + 2.3 + 8a =$

$37.2a + 2.3$

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

$\frac{8x^2}{4x}+3x=$

Video Solution

Step-by-Step Solution

Let's break down the fraction's numerator into an expression:

$8x^2=4\times2\times x\times x$

And now the expression will be:

$\frac{4\times2\times x\times x}{4x}+3x=$

Let's reduce and get:

$2x+3x=5x$

Answer

$5x$

Exercise #4

Simplifica la expresión:

$2x^3\cdot x^2-3x\cdot x^4+6x\cdot x^2-7x^3\cdot 5=$

Video Solution

Step-by-Step Solution

We'll use the law of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$ We'll apply this law to the expression in the problem:

$2x^3\cdot x^2-3x\cdot x^4+6x\cdot x^2-7x^3\cdot 5=2x^{3+2}-3x^{1+4}+6x^{1+2}-35x^3$ When we apply the above law to the first three terms from the left, while remembering that any number can always be considered as that number raised to the power of 1:

$a=a^1$ And in the last term we performed the numerical multiplication,

We'll continue and simplify the expression we got in the last step:

$2x^{3+2}-3x^{1+4}+6x^{1+2}-35x^3=2x^5-3x^5+6x^3-35x^3=-x^5-29x^3$ Where in the first stage we simplified the expressions in the exponents of the terms in the expression and in the second stage we combined like terms,

Therefore the correct answer is answer A.

Answer

$-x^5-29x^3$

Exercise #5

$\frac{9m}{3m^2}\times\frac{3m}{6}=$

Video Solution

Step-by-Step Solution

According to the laws of multiplication, we must first simplify everything into one exercise:

$\frac{9m\times3m}{3m^2\times6}=$

We will simplify and get:

$\frac{9m^2}{m^2\times6}=$

We will simplify and get:

$\frac{9}{6}=$

We will factor the expression into a multiplication:

$\frac{3\times3}{3\times2}=$

We will simplify and get:

$\frac{3}{2}=1.5$

Answer

$0.5m$

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