Lesson 5: Like Terms

New Concept

Learn to simplify expressions by combining like terms! This skill involves adding or subtracting terms with identical variable factors, making it easier to solve complex linear equations and inequalities.

What’s next

Next, you'll see this in action with worked examples, then apply it yourself in a series of practice cards and challenge problems.

Property

Two algebraic expressions are equivalent if they name the same number for all values of the variable.

Examples

• The expressions $4x + 2x$ and $6x$ are equivalent. If we test $x=3$, we get $4(3) + 2(3) = 12 + 6 = 18$, and $6(3) = 18$.
• The expressions $10-2x$ and $8x$ are not equivalent. If we test $x=1$, we get $10-2(1)=8$ and $8(1)=8$. But if we test $x=2$, we get $10-2(2)=6$ and $8(2)=16$. Since they are not equal for all values, they are not equivalent.
• The expressions $9y - 3y$ and $6y$ are equivalent. For any value of $y$, subtracting three $y$'s from nine $y$'s always results in six $y$'s.

Explanation

Think of equivalent expressions as two different ways to write the same value. No matter what number you substitute for the variable, they will always produce the same result because they are mathematically identical.

Property

Like terms are any terms that are exactly alike in their variable factors.
The numerical factor in a term is called the numerical coefficient, or just the coefficient of the term.

Examples

• In the expression $8a + 3a - b$, the terms $8a$ and $3a$ are like terms because they both have the variable factor $a$. The term $-b$ is not like them.
• The terms $5xy$ and $-2xy$ are like terms. The terms $5x$ and $-2xy$ are not like terms because their variable factors, $x$ and $xy$, are different.
• In the term $15z$, the number $15$ is the numerical coefficient. In a term like $x$, the coefficient is understood to be $1$. For a term like $-y$, the coefficient is $-1$.

Explanation

Like terms are terms that have the exact same variable part, including any exponents. You can think of them as being the same 'type' of item, which allows you to group them together through addition or subtraction.

Property

To add like terms: Add the numerical coefficients of the terms. Do not change the variable factors of the terms.
To subtract like terms: Subtract the numerical coefficients of the terms. Do not change the variable factors of the terms.
Replacing an expression by a simpler equivalent one is called simplifying the expression.

Examples

• To simplify $7b + 4b$, we add the coefficients: $(7+4)b = 11b$. We have combined the like terms into a single, simpler term.
• To simplify $12k - (-3k)$, we subtract the coefficients: $(12 - (-3))k = (12+3)k = 15k$. The variable factor $k$ remains unchanged.
• We cannot simplify $8p + 5q$ because $8p$ and $5q$ are not like terms. Their variable factors are different.

Explanation

When combining like terms, you only perform the operation on their coefficients. The variable part just tells you what 'family' of terms you're counting. It's like adding 4 apples and 5 apples to get 9 apples.

Property

Parentheses preceded by a plus sign may be omitted, and each term within parentheses keeps its original sign.
If an expression inside parentheses is preceded by a minus sign, we change the sign of each term within parentheses and then omit the parentheses and the minus sign.

Examples

• To simplify $(4a + 5) + (3a - 1)$, we can remove both sets of parentheses without changing signs: $4a + 5 + 3a - 1 = 7a + 4$.
• To simplify $(10x + 8) - (3x - 2)$, the minus sign flips the signs inside the second parentheses: $10x + 8 - 3x + 2 = 7x + 10$.
• In the expression $15 - (y + 6)$, we change the sign of each term inside the parentheses: $15 - y - 6 = 9 - y$.

Explanation

A plus sign before parentheses means you can just drop them. But a minus sign acts as a switch for every term inside—positive becomes negative, and negative becomes positive. This is like distributing a $-1$ to each term.

Property

1. Combine like terms on each side of the equation.
2. By adding or subtracting the same quantity on both sides of the equation, get all the variable terms on one side and all the constant terms on the other.
3. Divide both sides by the coefficient of the variable to obtain an equation of the form $x = a$.

Examples

• To solve $8x - 3x + 1 = 16$, first combine like terms: $5x + 1 = 16$. Then isolate $x$: $5x = 15$, so $x = 3$.
• To solve $7y - 5 = 3y + 11$, first gather variable terms on one side: $4y - 5 = 11$. Then gather constants: $4y = 16$. Finally, divide: $y = 4$.
• To solve $10z - (4z - 5) = 23$, first remove parentheses: $10z - 4z + 5 = 23$. Combine like terms: $6z + 5 = 23$. Isolate $z$: $6z = 18$, so $z=3$.

Explanation

First, simplify by cleaning up each side of the equation. Next, gather all your variable terms on one side and your numbers on the other. Finally, perform the last division to find the variable's value.