Like terms have exactly the same variables raised to exactly the same powers; only their coefficients may differ. 3x and 8x are like terms, and so are 5a2 and −2a2, but 4x and 4y are not. Only like terms can be combined by adding or subtracting their coefficients: 3x + 8x = 11x.
Sort the terms by their variable part, keeping each term’s sign, then add the coefficients in each group. Terms with different letters or different powers stay separate.
Formal definition and notation
Two terms of an expression are like terms when their variable parts are identical: the same variables, each with the same exponent (the order of the factors does not matter, so 2ab and −7ba are like terms). All constant terms are like terms of one another. By the distributive property, like terms combine as axn + bxn = (a + b)xn; unlike terms cannot be merged into a single term.
Symbol
Read it as
Meaning
3x, −x, 12x
three x, negative x, one half x
like terms: variable part x, coefficients 3, −1 and 12
All three terms have the variable part a, so they are like terms.
Combine the coefficients: 7 + 3 − 4 = 6.
Keep the variable part: 6a.
Answer:7a + 3a − 4a = 6a
Example 2: Several kinds of terms
Core
Simplify 5x2 − 2x + 3x2 + 8x − 6.
Group like terms, keeping their signs: (5x2 + 3x2) + (−2x + 8x) − 6.
5x2 + 3x2 = 8x2.
−2x + 8x = 6x.
The constant −6 has no partner, so it stays.
Answer:8x2 + 6x − 6
Example 3: Perimeter with like terms
Test-style
A rectangle has length2x + 3 and width x − 1. Write its perimeter in simplest form, then find the perimeter when x = 5.
Perimeter is the sum of all four sides: (2x + 3) + (x − 1) + (2x + 3) + (x − 1).
Combine the x-terms: 2x + x + 2x + x = 6x.
Combine the constants: 3 − 1 + 3 − 1 = 4. So P = 6x + 4.
For x = 5: 6(5) + 4 = 34.
Answer:P = 6x + 4; when x = 5, the perimeter is 34 units.
Common mistakes
✗ Mistake3x + 4y = 7xy
✓ Instead3x + 4y cannot be combined; it is already simplified.
Terms must have the same variables and exponents to be like terms. Check with numbers: x = 1, y = 1 gives 3 + 4 = 7, but x = 2, y = 1 gives 10, while 7xy = 14.
✗ Mistakex2 + x2 = x4
✓ Insteadx2 + x2 = 2x2
Adding like terms adds the coefficients (1 + 1 = 2); the exponent stays the same. Exponents add only when you multiply: x2 × x2 = x4.
✗ Mistake5x − 3 − 2x = 3x + 3
✓ Instead5x − 3 − 2x = 3x − 3
The minus sign in front of 3 belongs to the 3. When you regroup, each sign moves with its term.
✗ Mistakex + x = x2
✓ Insteadx + x = 1x + 1x = 2x
A lone x has coefficient 1. Adding two of them gives two x‘s, not x times x.
Combining like terms produces an equivalent expression: 2x + 5x and 7x are equal for every x. Equivalent expressions are the result; like terms are what you combine to get there.
Quick check: 3 questions
Question 1 of 3
Which pair are like terms?
Show answer and explanation
Answer: C.ab and ba are the same variable part, because the order of factors does not matter. Choice A has different variables, choice B different powers of x, and choice D a constant with an x-term. Matching coefficients do not make like terms.
Question 2 of 3
Simplify 9y − 4 + 2y + 10.
Show answer and explanation
Answer: B.9y + 2y = 11y and −4 + 10 = 6. 17y adds constants into the y-term, 11y − 14 gets the sign of 4 wrong, and 7y + 6 subtracts the y-terms.
Question 3 of 3
Simplify 3x2 + 2x − x2 + 4x.
Show answer and explanation
Answer: D.3x2 − x2 = 2x2 and 2x + 4x = 6x. 8x2 merges unlike terms, 4x2 adds instead of subtracting x2, and 2x4 changes the exponent.
Like Terms: frequently asked questions
How do you know if two terms are like terms?
Ignore the coefficients and compare what is left. If the letters match and each letter has the same exponent, they are like terms. 7m2n and −m2n are like terms; 7m2n and 7mn2 are not, because the exponents sit on different letters. Constants such as 4 and −9 are always like terms.
Why can't you combine unlike terms?
Because they stand for different kinds of quantities. If x is the number of pens and y the number of notebooks, 3x + 4y is 3 pens and 4 notebooks, not 7 of anything. A quick number test shows it: with x = 2 and y = 5, 3x + 4y = 26, but 7xy = 70.
What does combining like terms mean?
It means adding or subtracting the coefficients of like terms to write a shorter, equivalent expression. In 4n + 3 + 2n − 1, the n-terms give 6n and the constants give 2, so the expression becomes 6n + 2. The value stays the same for every n; only the form is simpler.
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