The Anatomy of an Expression
An expression is a sum of terms, and only like terms — those sharing the exact same variable part — may be combined by adding their coefficients. · 9 min
Last folio, one letter held one number. Real expressions run longer: 3x + 5 + 2x, or 7a − 4 + a. Before you can work with an expression, you need to see its parts — because some parts can merge and some must keep their distance, and telling those apart is this folio's whole job.
Guess before you learn
Simplify 3x + 5 + 2x as far as it will go. What do you get?
The two x-terms pool into 5x, but the lone 5 is a different kind of part and stays separate: 5x + 5. If you guessed 10x, keep the pencil mark — folding the constant into the x's is the most common first instinct, and the next section shows exactly why it changes the value.
9–12
3–5
Say x is the number of stickers in one pack. Then 3x means three packs, and 2x means two more packs. Together that is five packs: 5x. The 5 loose stickers are not packs, so the total is 5x + 5 — five packs and five singles.
You may only add pieces that count the same kind of thing.
6–8
An expression is a sum of terms — the pieces separated by + and − signs. In 3x + 5 + 2x the terms are 3x, 5, and 2x. Each term is a coefficient (the number in front) times a variable part, or a bare number called a constant.
Like terms share the exact same variable part. 3x and 2x are like; 3x and 5 are not; neither are 3x and 3x². Combine like terms by adding their coefficients: 3x + 2x = 5x. So 3x + 5 + 2x becomes 5x + 5 — and there it stops.
9–12
'Exact same variable part' means the same letters raised to the same powers. 2x and 7x qualify. 2x and 2x² do not — at x = 3 they equal 6 and 18, different numbers, so merging them would change the expression's value. Order inside a term is harmless: 4xy and −2yx are like terms, because multiplication commutes.
Combining is the distributive property read backward: 3x + 2x = (3 + 2)x = 5x. That is the entire license. It is also testable — a correct simplification agrees with the original at every value of x, so one substitution check can expose a false one.
K–2
You have 3 red blocks and 2 blue blocks. A friend hands you 2 more red blocks. Now: 5 red, 2 blue. Red joins red. Blue joins blue.
Letters work the same way. Three x's and two more x's make five x's. A plain number is not an x, so it stays in its own pile.
Undergrad
Treat expressions as polynomials in ℝ[x]. The monomials 1, x, x², … form a basis, and every polynomial is a unique linear combination of them. 'Like terms' are scalar multiples of the same basis element; collecting them is coordinate-wise addition. 5x + 5 resists because its terms sit on different basis elements — it factors as 5(x + 1) but cannot shorten to one term.
Uniqueness of representation is the deep fact: two polynomials over ℝ agree as functions exactly when their coefficients match term by term. Simplification is therefore canonical — everyone who combines correctly lands on the same normal form.
Postgrad
In the free commutative algebra R[x₁, …, xₙ], every element has a unique normal form: a finite linear combination of distinct monomials. 'Combining like terms' is a rewriting system — merge equal monomials, drop zero coefficients — that is terminating and confluent, so any order of moves reaches the same canonical form.
That normal form is what makes equality of polynomial expressions decidable, and imposing a monomial order on it is the first step toward Gröbner bases. The schoolroom pile-sorting is the algebra's grading, met early.
like terms
Terms whose variable parts match exactly — same letters, same powers. 3x and 2x are like terms; 3x and 3x² are not.
Why is adding coefficients legal at all? Because of the distributive property, run in reverse. 3x + 2x means three x's plus two more x's — that is (3 + 2)x, which is 5x. No such move exists for 5x + 5: the 5 has no x to pool with. The test of any simplification is blunt: the new expression must equal the old one at every value of x. Try it below.
Simplify 6a + 4 − 2a + 3 — the steps fade as you master them
(6a − 2a) + (4 + 3)
4a + (4 + 3)
4a + 7
Why is this true?
Why would merging 5x and 5 into 10x change the expression?
Because 5x + 5 and 10x disagree at almost every input — at x = 2 they give 15 and 20. A constant adds the same amount regardless of x, while an x-term grows with x, so they carry different jobs and cannot trade places.
So an expression is a sum of terms, each with a coefficient and a variable part — and only matching variable parts may pool. One habit before you go: when a lone letter appears, read its silent coefficient of 1, so x + 4x is 5x, not 4x. Next folio: what happens when an expression contains several operations at once, and why everyone must read it in the same order.
Practice — new ink and old, interleaved
1.Evaluate 3n + 2 when n = 7.
2.Without looking back: what is a variable, and what does 7m mean?
A variable is a letter standing for a number not yet fixed; 7m means 7 × m.
How close were you? Grade yourself honestly — it sets your review date.
3.Which expression is fully simplified?
4.Without looking back: what are like terms, and how do you combine them?
Like terms share the exact same variable part; you combine them by adding their coefficients, so 3x + 2x = 5x while 5x + 5 stays as it is.
How close were you? Grade yourself honestly — it sets your review date.
5.Put the steps for simplifying 5y + 3 + 2y − 1 in order.
- Group like terms: (5y + 2y) + (3 − 1)
- Add the coefficients of the y-terms: 7y
- Combine the constants: 2
- Write the result: 7y + 2
6.Simplify 7t + 2 − 4t, then evaluate it at t = 3.
7.A pencil costs 2 dollars. In one sentence with an expression in it: what do p pencils cost, and why?