Lesson 18: Combining Like Terms

New Concept

Algebra is a powerful mathematical language that uses variables to represent unknown quantities and express relationships between them.

What’s next

We begin with a core skill: simplifying expressions. Next, you’ll work through examples of combining like terms, a key rule in algebraic manipulation.

Property

Two or more terms that have the same variable or variables raised to the same power are like terms. Terms with different variables or the same variables raised to different powers are unlike terms.

Examples

• $7x^2$ and $3x^2$ are like terms because they share the same variable family, $x^2$.
• $7y^2$ and $3x^2$ are unlike terms because their variables, $y$ and $x$, are different.
• $7y^4$ and $3y^2$ are unlike terms because their exponents are different, making them distant cousins, not twins.

Explanation

Think of variables and their exponents as a family name. You can only combine terms from the exact same family! The numbers in front, called coefficients, just tell you how many members of that family you have. For terms to be alike, their variable parts must be identical twins.

Property

To combine like terms, use the Distributive Property in reverse: $ax + bx = (a+b)x$.

Examples

• $6x + 9x = (6 + 9)x = 15x$
• $-5y - (-2y) + 4y = (-5 + 2 + 4)y = 1y = y$
• $7ab - 2c + 5ba = 7ab + 5ab - 2c = (7+5)ab - 2c = 12ab - 2c$

Explanation

Time to organize the party! Round up all the terms that are alike and group them together. Then, simply add or subtract their coefficients. The variable part is just a label that comes along for the ride and doesn't change. It’s like saying '5 apples plus 7 apples equals 12 apples,' not '12 super-apples!'

Property

To combine like terms with exponents, the variable and its power must match exactly: $ax^n + bx^n = (a+b)x^n$.

Examples

• $x^4 + y^2 + x^4 + y^2 = (1+1)x^4 + (1+1)y^2 = 2x^4 + 2y^2$
• $4m^3 - 3m^3 + 5m^3 = (4 - 3 + 5)m^3 = 6m^3$
• $3x^2y^4 - 9y^4x^2 + xy = (3-9)x^2y^4 + xy = -6x^2y^4 + xy$

Explanation

Welcome to the big leagues! The rule is the same, but now the 'family name' includes the exponent. A term like $k^3$ is in a completely different family from $k^2$. You can only combine terms if their variable and exponent are identical. Remember, when you combine them, the exponent does not change!