Identify like terms by matching variables and exponents, then combine them to simplify algebraic expressions. Apply the distributive property before combining like terms in multi-step expressions.
A lesson of Algebra 1, in Foundations of Algebra, about 30 minutes.
By the end of this lesson, you will be able to identify like terms in any algebraic expression, combine them by adding or subtracting their coefficients, and simplify multi-step expressions that require distributing before combining. with complete confidence in handling negative coefficients.
You Can Only Add Things of the Same Kind
Simplifying an expression means writing it in its shortest equivalent form. The
operation that does most of the work is combining like terms: adding
together the terms that represent the same kind of thing.
You have been doing this with numbers your whole life.
That last one is important: subtracting identical terms gives zero, and the terms disappear from the expression entirely.
Example 1: Two Groups and Nothing Else
Here is the whole method on one expression, worked all the way through.
Simplify $6y + 2 - 4y + 7$
$6y$ and $-4y$ are like terms, because both count $y$. The $2$ and the $7$ are
like terms too, because both are plain numbers. So there are two groups here,
and each one gets added up on its own.
Move the like terms next to each other. Nothing has been added yet, only rearranged. 6y - 4y + 2 + 7
Add the y-terms. The constants come along unchanged. 2y + 2 + 7
Add the constants 2y + 9
Worked Example 1: Three Groups at Once
Simplify $8x + 3y - 5x + 2y - 1$
Three groups this time: the $x$-terms, the $y$-terms, and a lone constant.
Example 2: When a Letter Stands Alone
A term with no number in front of it still has one. It is a 1, and it is
invisible.
Simplify $7b - b$
Write the invisible 1 in, so both coefficients are visible 7b - 1b
Subtract the coefficients 6b
Worked Example 2: The Invisible Coefficient
Simplify $4a^2 - 3a + 7 - a^2 + 5a - 2$
The trap here is $-a^2$. It has a coefficient even though you cannot see it.
Example 3: Brackets Come First
You cannot combine anything while it is still trapped inside brackets.
Simplify $4(x + 2) + 3x$
Multiply everything inside the bracket by 4, keeping the rest of the expression 4x + 8 + 3x
Add the x-terms 7x + 8
Worked Example 3: Distribute Before You Combine
Simplify $3(2x - 4) + 5x - 1$
When parentheses are present, you must distribute before combining.
Worked Example 4: Subtracting a Whole Bracket
Simplify $2(3x + 1) - 4(x - 5)$
Two sets of brackets, and the second one is being subtracted. That minus sign
is the whole difficulty of this question.
Common Mistakes
Mistake 1: Combining unlike terms
Wrong: $3x + 5x^2 = 8x^2$ or $8x^3$
Right: $3x + 5x^2$ cannot be simplified. They are unlike terms
$x$ (first power) and $x^2$ (second power) are as different as apples and oranges. You cannot add them.
Mistake 2: Forgetting that a variable with no written coefficient has coefficient 1
Wrong: $5a - a = 5$
Right: $5a - a = 5a - 1a = 4a$
$a$ means $1a$. The coefficient is $1$, not $0$. So $5a - a = (5-1)a = 4a$.
Mistake 3: Combining coefficients but changing the variable part
Wrong: $4x^2 + 3x^2 = 7x^4$
Right: $4x^2 + 3x^2 = 7x^2$
The exponent stays the same. You add the coefficients, not the exponents.
Mistake 4: Not distributing the negative to all terms before combining
Wrong: $5 - 2(x - 3) = 5 - 2x - 3 = 2 - 2x$
Right: $5 - 2(x - 3) = 5 - 2x + 6 = 11 - 2x$
$-2 \times (-3) = +6$. The negative outside must be distributed to every term inside.
Mistake 5: Combining terms separated by multiplication
Wrong: $3x \cdot 5x = 8x$
Right: $3x \cdot 5x = 15x^2$
Combining like terms applies to addition and subtraction, not multiplication. When terms are multiplied, multiply the coefficients and add the exponents.
Where This Goes Next
Expressions built from terms with whole-number exponents on variables are called polynomials. The expression $4a^2 + 2a + 5$ from Example 2 is a polynomial in the variable $a$.
When writing a polynomial in standard form, arrange the terms in descending order of degree (highest exponent first):
The degree of a polynomial is the highest exponent on any variable term. $4a^2 + 2a + 5$ has degree $2$, it's a quadratic. Degree 1 is linear. Degree 3 is cubic.
This vocabulary matters because different degrees of polynomials behave differently: linear ones give straight lines when graphed, quadratics give parabolas, and so on. Simplifying to standard form first makes everything that follows. graphing, solving, analyzing. much cleaner.
Check Your Understanding
Four quick ones. No steps given this time.
Summary
Like terms share the same variable(s) raised to the same exponent(s).
To combine: add or subtract the coefficients; the variable part stays identical.
$$7x^2 - 3x^2 = (7-3)x^2 = 4x^2$$
The full simplification routine:
Distribute to clear all parentheses
Identify groups of like terms
Combine each group by adding/subtracting coefficients
Write the result in standard form (descending degree)
Quick checks:
Different exponents → cannot combine ($3x \neq 3x^2$)
Different variables → cannot combine ($3x \neq 3y$)
No written coefficient → coefficient is $1$ (so $-x = -1x$)
Distributing a negative flips every sign inside the parentheses