The Central Idea
Suppose you have three apples and then receive two more apples. You now have five apples.
Algebra uses the same idea:
\[ 3x+2x=5x. \]The two terms can be combined because both represent multiples of the same quantity \(x\).
Combining like terms does not change the mathematical value of an expression. It replaces several compatible terms with one equivalent term.
What Is a Term?
A term is a part of an expression separated from other terms by addition or subtraction at the outermost level.
For example, in
\[ 5x^2-3x+7, \]the terms are
\[ 5x^2,\qquad -3x,\qquad 7. \]Each term may contain numbers, variables, powers, or products of several factors.
Coefficient and Variable Part
A variable term can often be viewed as two parts:
\[ \text{coefficient}\times\text{variable part}. \]For example, in
\[ 7x^2y, \]the coefficient is
\[ 7 \]and the variable part is
\[ x^2y. \]In
\[ -3ab^2, \]the coefficient is
\[ -3 \]and the variable part is
\[ ab^2. \]This separation helps us identify like terms.
What Makes Terms Like Terms?
Like terms have exactly the same variable part.
For example,
\[ 4x \quad\text{and}\quad -9x \]are like terms because both have variable part \(x\).
Likewise,
\[ 3x^2y \quad\text{and}\quad 11x^2y \]are like terms because both have variable part \(x^2y\).
The coefficients may differ. The variable factors and their exponents must match.
Examples of Like and Unlike Terms
| Terms | Like? | Reason |
|---|---|---|
| \(3x\) and \(8x\) | Yes | Both have variable part \(x\). |
| \(5x^2\) and \(-2x^2\) | Yes | Both have variable part \(x^2\). |
| \(4xy\) and \(9xy\) | Yes | Both have variable part \(xy\). |
| \(x\) and \(x^2\) | No | The exponents of \(x\) differ. |
| \(3xy\) and \(3x\) | No | One term contains \(y\) and the other does not. |
| \(2a^2b\) and \(2ab^2\) | No | The exponents are attached to different variables. |
Constants Are Like Terms
Numbers without variables are called constant terms.
All constant terms are like terms with one another.
For example,
\[ 7-3+10=14. \]In the expression
\[ 4x+7-2x+5, \]the variable terms \(4x\) and \(-2x\) are like terms, while the constants \(7\) and \(5\) are another group of like terms.
How Combining Like Terms Works
When terms are alike, their variable part can be treated as a common unit.
Consider
\[ 3x+5x. \]This means
\[ 3(x)+5(x). \]Because both terms contain the same factor \(x\), their coefficients can be added:
\[ (3+5)x=8x. \]Therefore,
\[ 3x+5x=8x. \]Combining like terms is therefore a form of collecting common algebraic units.
Why the Rule Is Valid
The transformation
\[ 3x+5x=8x \]is justified by the distributive property:
\[ 3x+5x=(3+5)x. \]Then
\[ (3+5)x=8x. \]So the familiar instruction “add the coefficients” is not an isolated rule. It is a compact use of a more general algebraic law.
The same structure works with subtraction:
\[ 9x-4x=(9-4)x=5x. \]Combining Positive and Negative Coefficients
The coefficients of like terms may be positive or negative.
For example,
\[ 7x-10x=-3x. \]We combine the coefficients:
\[ (7-10)x=-3x. \]Likewise,
\[ -4y-6y=-10y. \]and
\[ -8a+3a=-5a. \]The sign belongs to the coefficient.
Invisible Coefficients
A variable written without an explicit numerical coefficient has coefficient \(1\):
\[ x=1x. \]A negative variable written without an explicit numerical coefficient has coefficient \(-1\):
\[ -x=-1x. \]Therefore,
\[ x+4x=5x, \]and
\[ 6x-x=5x. \]Similarly,
\[ x-x=0x=0. \]Combining Several Groups of Terms
An expression may contain several different families of like terms.
Consider
\[ 4x+3y-2x+5y. \]Group the \(x\)-terms and \(y\)-terms mentally:
\[ (4x-2x)+(3y+5y). \]Then combine each group:
\[ 2x+8y. \]Thus,
\[ 4x+3y-2x+5y=2x+8y. \]Combining Variables and Constants
Consider
\[ 6x+4-2x+9. \]The \(x\)-terms combine:
\[ 6x-2x=4x. \]The constants combine:
\[ 4+9=13. \]Therefore,
\[ 6x+4-2x+9=4x+13. \]The \(x\)-terms cannot be combined with the constants because they represent different algebraic kinds of quantities.
Why Unlike Terms Cannot Be Combined
Consider
\[ 3x+4y. \]There is no general way to rewrite this as
\[ 7xy, \] \[ 7x, \]or any other single ordinary term while preserving its value for all \(x\) and \(y\).
For example, if
\[ x=2,\qquad y=3, \]then
\[ 3x+4y=3(2)+4(3)=18. \]But
\[ 7xy=7(2)(3)=42. \]The expressions are not equivalent.
Terms can be combined only when their algebraic structure supports the replacement.
Exponents Must Match
The expressions
\[ x \]and
\[ x^2 \]are not like terms.
Therefore,
\[ 3x+5x^2 \]cannot be combined into a single term by adding the coefficients.
Even though both terms contain \(x\), they represent different powers of \(x\).
For example, when \(x=2\),
\[ 3x+5x^2=6+20=26. \]But the incorrect expression
\[ 8x^2 \]would equal
\[ 32. \]The mismatch shows why exponent structure matters.
Order of Variables Does Not Matter in a Product
Because multiplication of real numbers is commutative,
\[ xy=yx. \]Therefore,
\[ 3xy+5yx \]contains like terms.
We may rewrite \(yx\) as \(xy\):
\[ 3xy+5xy=8xy. \]Likewise,
\[ 2ab^2+7b^2a=9ab^2. \]Before deciding whether terms are alike, it can be useful to place variable factors into a consistent order.
Like Terms with More Than One Variable
The same principle extends naturally to products of several variables.
Consider
\[ 4x^2y-9x^2y+3x^2y. \]All three terms have variable part
\[ x^2y. \]So we combine the coefficients:
\[ (4-9+3)x^2y=-2x^2y. \]Therefore,
\[ 4x^2y-9x^2y+3x^2y=-2x^2y. \]Parentheses Can Hide Like Terms
Sometimes like terms are not yet visible because part of the expression is grouped.
Consider
\[ 3x+(2x+5). \]Because the outer addition does not change the signs inside the parentheses, we may read this as
\[ 3x+2x+5. \]Then
\[ 3x+2x=5x, \]so
\[ 3x+(2x+5)=5x+5. \]More complicated removal of parentheses depends on distribution and sign handling, which will be developed in the next lesson.
Combining Like Terms Does Not Solve an Equation by Itself
Combining like terms simplifies expressions that appear inside equations, but it is not the same thing as solving the equation.
Consider
\[ 3x+4x+2=23. \]First combine like terms:
\[ 7x+2=23. \]This simplifies the equation, but \(x\) is not isolated yet.
Continue using the properties of equality:
\[ 7x=21 \]and
\[ x=3. \]Combining like terms is one transformation within a larger solution process.
Combining Terms Across an Equal Sign
Terms on opposite sides of an equation are not combined directly.
For example, in
\[ 3x+2=5x-6, \]the terms \(3x\) and \(5x\) are like terms, but they are on different sides of the equality.
To collect them on one side, we first use a property of equality. Subtract \(3x\) from both sides:
\[ 3x+2-3x=5x-6-3x. \]Then
\[ 2=2x-6. \]Only after the terms are on the same side can they be combined within that expression.
A Useful Organizational Strategy
For longer expressions, it helps to identify families of like terms before doing any arithmetic.
Consider
\[ 5x^2-3x+7+2x^2+8x-4. \]The expression contains three families:
- \(x^2\)-terms: \(5x^2\) and \(2x^2\);
- \(x\)-terms: \(-3x\) and \(8x\);
- constants: \(7\) and \(-4\).
Combine within each family:
\[ 7x^2+5x+3. \]This organization becomes increasingly useful as expressions become larger.
Worked Example: Several Variables
Simplify
\[ 7a-3b+4a+8b-2. \]Combine the \(a\)-terms:
\[ 7a+4a=11a. \]Combine the \(b\)-terms:
\[ -3b+8b=5b. \]The constant remains \(-2\).
Therefore,
\[ 7a-3b+4a+8b-2=11a+5b-2. \]Worked Example: Powers
Simplify
\[ 6x^3+2x-4x^3+9x. \]The cubic terms combine:
\[ 6x^3-4x^3=2x^3. \]The linear terms combine:
\[ 2x+9x=11x. \]Therefore,
\[ 6x^3+2x-4x^3+9x=2x^3+11x. \]The terms \(2x^3\) and \(11x\) cannot be combined because their variable parts differ.
Worked Example: Checking by Substitution
Suppose we simplify
\[ 4x+7x-3 \]to
\[ 11x-3. \]We can check the equivalence by choosing a value, such as \(x=2\).
Original expression:
\[ 4(2)+7(2)-3=8+14-3=19. \]Simplified expression:
\[ 11(2)-3=22-3=19. \]The matching values support the simplification.
A numerical check is useful for catching mistakes, although checking one value alone does not prove two expressions are equivalent for every possible value. The algebraic justification comes from the distributive property and the laws of arithmetic.
Common Errors
Adding Exponents Instead of Coefficients
An incorrect simplification is
\[ 3x^2+4x^2=7x^4. \]When adding like terms, the variable part stays the same:
\[ 3x^2+4x^2=7x^2. \]Exponents are added in certain multiplication rules, not when adding like terms.
Combining Unlike Terms
Another incorrect simplification is
\[ 2x+3=5x. \]The terms \(2x\) and \(3\) are unlike, so they cannot be combined.
Losing a Negative Sign
In
\[ 8x-11x, \]the second coefficient is \(-11\), so
\[ 8x-11x=(8-11)x=-3x. \]Signs are part of the coefficients and must be carried with the terms.
Practice
1. Identify the Like Terms
Which terms are like in
\[ 4x^2+7x-3x^2+5-2x? \]Check Answer
The like-term groups are:
\[ 4x^2 \text{ and } -3x^2, \] \[ 7x \text{ and } -2x, \]and the constant \(5\) stands alone.
2. Combine the Terms
Simplify:
\[ 9x-4x+6. \]Check Answer
\[ 9x-4x+6=5x+6. \]3. Several Variables
Simplify:
\[ 3a+5b-7a+2b. \]Check Answer
\[ 3a-7a+5b+2b=-4a+7b. \]4. Powers
Simplify:
\[ 8x^2+3x-5x^2+6x. \]Check Answer
\[ 8x^2-5x^2+3x+6x=3x^2+9x. \]5. Decide Whether the Terms Are Like
Can
\[ 4x^2y+9xy^2 \]be combined into one term by adding the coefficients?
Check Answer
No. The variable parts \(x^2y\) and \(xy^2\) are different, so the terms are not like.
Key Ideas
- Like terms have exactly the same variable part.
- Coefficients may differ while the variable factors and exponents remain the same.
- Constant terms are like terms with other constants.
- Combining like terms adds or subtracts coefficients while leaving the common variable part unchanged.
- The distributive property explains why combining coefficients is valid.
- Unlike terms cannot generally be merged into a single ordinary term.
- Signs belong to coefficients and must be preserved.
- Combining like terms simplifies expressions but does not by itself solve an equation.
- Reordering terms can help reveal groups of like terms without changing the expression's value.
Where This Leads
Combining like terms lets us collect pieces that already share the same algebraic structure.
Next, we will study Distribution, Collection, and Rearrangement, which gives us broader tools for exposing like terms, removing or introducing grouping, and reorganizing expressions into more useful equivalent forms.