Chapter 3 · Algebraic foundations
Combine Like Terms, Simplify, and Find Errors
Learn to combine like terms, simplify linear expressions by distributing and collecting, and find and fix the step where a simplification goes wrong.
An expression combines numbers and letters with operations. Each letter is a variable: it stands for a number, and in this lesson it may be any real number (any number on the number line). For example, is an expression with the variable .
You can often rewrite a long expression in a shorter form that has exactly the same value. Here you will learn how to do that, and how to catch the step where a rewrite goes wrong.
Three more words help you talk about expressions. The expression has three terms, the parts you add: , and . In the term , the number is the coefficient, the number that multiplies the variable. A term with no variable, such as , is a constant.
What does mean? The term means , and means . Together that makes eight copies of , so has the same value as for any number you choose.
You can collect and because they are like terms: terms with exactly the same variable part. Constants have no variable part, so constants are like terms with each other.
| Terms | Like terms? | Reason |
|---|---|---|
and | Yes | Both are a number times . |
and | Yes | means , so both are a number times . |
and | Yes | Both are constants. |
and | No | has the variable , but has no variable. |
and | No | The variables and are different. |
Why can you collect like terms? Two expressions are equivalent when they have the same value for every value their variables are allowed to take. In this lesson that means every real number.
You already know the distributive property. With the shared factor written on the right, it says for all real numbers , and . Read from right to left, it pulls a shared factor (something that multiplies every term) out of a sum. In the shared factor is :
The distributive property holds for all real numbers, so and are equivalent. They agree for every value of , not only for the values you happen to try.
Subtraction works the same way, because subtracting a term is the same as adding its opposite (the same term with the other sign):
This is why the sign in front of a term belongs to that term. In the terms are and .
Key concept
Combine like terms by adding their coefficients
To combine like terms, add their coefficients and keep the variable part. Constants combine by ordinary addition. The distributive property is the reason, so the rule is true for all real numbers , and . The new expression is equivalent to the old one: both have the same value for every value the variable may take. To subtract a term, add its opposite. Terms that are not like terms have no shared variable part to pull out, so they stay separate.
Worked example
Add and subtract like terms
Simplify , where can be any real number.
Rewrite each subtraction as adding the opposite, so every term carries its own sign. Also write x as 1x to show its coefficient. This changes only how the expression is written, not its value.
Move like terms next to each other. Everything is now addition, and the commutative and associative properties of addition say a sum keeps its value in any order and grouping.
Combine the x terms by adding their coefficients. The distributive property makes this true for every value of x.
Combine the constants: 5 plus negative 9 is negative 4.
Join the two results. Adding negative 4 is the same as subtracting 4. The terms 6x and negative 4 are not like terms, so you are finished.
Answer
The expression simplifies to . The two expressions have the same value for every real number .
A linear expression is one that, after you multiply out any parentheses, is a sum of terms where each term is a number alone or a number times one variable. Examples are and .
To simplify a linear expression, rewrite it as an equivalent expression with no parentheses and no like terms left to combine. A reliable plan has two moves, and you already know why each one is valid:
- Distribute to remove parentheses. The outside factor, with its sign, multiplies every term inside.
- Combine like terms by adding their coefficients.
Each move replaces the expression with an equivalent one. So the simplified result has the same value as the original for every value of the variable.
Worked example
Distribute, then combine
Simplify , where can be any real number.
Distribute the 4 to both terms inside the parentheses. Read 2x minus 3 as 2x plus negative 3, so the second product is 4 times negative 3. The distributive property keeps the value the same for every x.
Replace the product with the equal expression you just found. The other terms stay as they are.
Group the like terms. Subtracting 12 is adding negative 12, so the properties of addition allow the new order.
Add the coefficients of the x terms, and add the constants.
Write adding negative 5 as subtracting 5. No parentheses and no like terms are left, so the expression is simplified.
Answer
The expression simplifies to . The two expressions are equal for every real number .
Worked example
Distribute a fraction and a negative factor
Simplify , where can be any real number.
Distribute one half to both terms of the first group. Half of 8x is 4x, and half of 6 is 3.
Subtracting 3 times a group is the same as adding negative 3 times that group, so distribute negative 3. Negative 3 times 2x is negative 6x. Negative 3 times negative 4 is positive 12, because the product of two negative numbers is positive.
Add the two results. Adding negative 6x is the same as subtracting 6x.
Group the like terms, keeping the sign in front of each term with that term.
Add the coefficients of the x terms, and add the constants.
Answer
The expression simplifies to . The two expressions are equal for every real number .
Now use the same ideas to check work, whether it is yours or someone else's. In a correct simplification, every line is equivalent to the line before it. An error is a step where that stops being true. You have two tools for finding it:
- Reason with the properties. For each step, ask which property allows it, and whether the step applies it to every term with the correct sign.
- Test a number. Choose a value the variable is allowed to take and work out the value of every line. If one line and the next give different values, the step between them is not valid. Matching values are a good sign but not a proof, so check the reasoning too. A value such as or works well. At every term becomes , so the test cannot see a mistake in those terms.
When you find the faulty step, correct that line and redo every line after it, because the later lines depend on the mistake.
| Step | Question to ask | Why it matters |
|---|---|---|
Distribute | Did the outside factor, with its sign, multiply every term inside? | The distributive property applies the factor to every term. A minus sign before a group acts as a factor of . |
Regroup | Did every term keep the sign in front of it? | You can reorder addition freely but not subtraction, so a subtracted term moves as a negative term. |
Combine | Does the step combine only like terms and add their coefficients correctly? | Adding coefficients uses the distributive property, which needs the same variable part in each term. |
Worked example
Find and correct an error
A student simplified , where can be any real number. Line 1: . Line 2: . Line 3: . Find the error and give the correct simplification.
Choose a test value and work out the value of the original expression. Any real number is allowed, so try x = 2.
Work out the value of the student's lines at x = 2. Line 1 gives 18, the same as the original, and its distribution is correct: 3 times 2x is 6x, and 3 times 1 is 3. Line 2 gives 8. The value changes between Line 1 and Line 2, so that step is not valid.
Use the properties to see what went wrong. Subtracting a group is the same as adding negative 1 times the group, so negative 1 must multiply both terms. The student applied the minus sign to x but left negative 5 unchanged.
Correct Line 2 by putting the equal expression from the last step in place of the subtracted group.
Line 3 depends on the faulty line, so redo it. Group the like terms with their signs, then add the coefficients of the x terms and add the constants.
Test the new result at x = 2. It gives 18, which matches the original. The match supports the answer, and the properties used in each step are what guarantee it for every value of x.
Answer
The error is in Line 2: the student applied the minus sign to but not to . The correct simplification is , which equals for every real number .
Takeaway. Simplifying changes how an expression is written, never what it is worth. Distribute to remove parentheses, then combine like terms by adding their coefficients. A property that holds for all real numbers backs each step, so the simplified expression equals the original for every value of the variable. When a result looks wrong, find the first line that is not equivalent to the line before it, fix that line, and redo the lines after it. Next, you will factor linear expressions and identify when two linear expressions are equivalent.