Combining Like Terms: Three Steps, Why It Works, and Worked Examples

Why can the cashier merge those lines? Because the items scanned are the same kind of goods: the counts add, and the item name rides along unchanged. Combining like terms copies that move.

An algebra expression is messier than a receipt: 5x, 3, x², 4x, −2, and 2x² all jumbled together. Before combining like terms, send each term home. The figure below shows those six terms sorted into piles.

With the piles set, every combining like terms problem reduces to three steps:

1. Identify the like terms. Same letters, same exponent on every letter — that is one family. Plain numbers form their own family.
2. Regroup. Use the commutative property of addition to move family members together — the sign travels with each term.
3. Add the coefficients. Add only the numbers in front; copy the letter part exactly as it was — the whole arithmetic of combining like terms.

Run the smallest combining like terms case. Three x's plus five x's make eight x's: $3x + 5x = (3+5)x = 8x$.

In letters it reads $ax + bx = (a+b)x$ — the one-line heart of combining like terms. Substitute $x = 10$ as a check: the left side gives $3\times10 + 5\times10 = 80$, and the right side gives $8\times10 = 80$. Equal, so the merge lost nothing. Sort, regroup, add the coefficients — combining like terms in one breath.

An 1896 classic states it crisply. Hall and Knight's Elementary Algebra: "Like terms, or similar terms, do not differ, or differ only in their numerical coefficients." Either identical, or different only in the number out front — that is the membership rule for every combining like terms decision.

The test looks only at the variable part: same letters, and the same exponent on each letter.

Two reminders for combining like terms: the coefficient of x is 1 ($x = 1 \cdot x$), and the coefficient of −x is −1. The hidden 1 usually stays unwritten — but the arithmetic of combining like terms must still count it.

Why may $3x + 5x$ merge while $3x + 2x^2$ may not? The distributive law stands behind all combining like terms.

Factor the common $x$ out of $3x + 5x$: $3x + 5x = (3+5)x = 8x$. That is the distributive law run in reverse — distributing gives $3x$, factoring returns $3x$, and the two directions lock the equality shut. This is the license that combining like terms operates on.

The figure draws the same combining like terms fact. Three segments of x joined to five segments of x make one strip of eight. But three segments of x beside two blocks of x² do not splice — different length, different thickness, no single bar. Combining like terms can only join what matches.

Let numbers talk — substitution is the built-in lie detector of combining like terms. Suppose someone forces $3x + 2x^2$ into $5x^3$. Test at $x = 2$: the left side gives $3\times2 + 2\times4 = 14$, the right side $5\times8 = 40$. Not equal — the fake merge collapses on the spot. Test the real combining like terms merge, $3x + 5x = 8x$, at $x = 10$: left 80, right 80. Exact.

Combining like terms also pays in work saved. Durell's 1911 School Algebra works one example: simplify $3ab^2 + 5ab^2 + 2ab^2$ with $a = 2$ and $b = 3$. Substituting directly means three full computations: $54 + 90 + 36 = 180$. Combining like terms first gives $10ab^2$, and one step finishes: $10\times2\times9 = 180$. In the book's words, more than half the work is saved.

For unlike terms the old texts leave one sentence: the addition of dissimilar terms can only be indicated, never performed. So $3x + 2x^2$ is already in simplest form — combining like terms stops there.

Problem. Simplify $8x + 5x + x + 2x + 6x$ — a one-family combining like terms drill.

Solution. Every term carries the letter part $x$: one family. The third term writes no coefficient. Its coefficient is the hidden 1, since $x = 1x$ — the detail that combining like terms quietly relies on.

Add the coefficients: $8 + 5 + 1 + 2 + 6 = 22$.

Answer. $8x + 5x + x + 2x + 6x = 22x$ — combining like terms kept it to one term.

A row of negatives runs the same way: $-3x - 5x - 7x - x$. The coefficient is $-(3+5+7+1) = -16$, so the sum is $-16x$. Signs never change who counts as like — in combining like terms they only ride along into the addition.

Problem. Simplify $9a - 5a + 4a - 2a$ — combining like terms with mixed signs.

Solution. All four terms contain $a$: like terms, but the signs differ — the safe combining like terms habit is to split into two piles:

• Positive pile: $9a + 4a = 13a$;
• Negative pile: $-5a - 2a = -7a$.

Inside $13a$ there is exactly $7a$ to cancel against $-7a$ — quantities equal in size and opposite in sign sum to 0. What remains: $13a - 7a = 6a$.

Answer. $9a - 5a + 4a - 2a = 6a$.

Check at $a = 1$: the left side gives $9 - 5 + 4 - 2 = 6$, and the right side gives $6 \times 1 = 6$. Agreed — this combining like terms result survives the substitution test.

Problem. Add $(4x^2 + 3x + 2) + (3x^2 - 4x - 3) + (-2x^2 - x - 5)$ — full-width combining like terms.

Solution. Line up like terms in columns; combine each column on its own:

The columns: $4 + 3 - 2 = 5$; $3 - 4 - 1 = -2$; $2 - 3 - 5 = -6$.

Answer. $5x^2 - 2x - 6$.

Check (substitution, cheapest at $x = 1$): first expression $4 + 3 + 2 = 9$; second $3 - 4 - 3 = -4$; third $-2 - 1 - 5 = -8$. The rows total $9 - 4 - 8 = -3$, and the answer expression gives $5 - 2 - 6 = -3$. Agreed — a cheap audit for any combining like terms work.

One more phenomenon from this combining like terms table: if a column's coefficients happen to total 0, that whole term vanishes from the result.

1. Forcing unlike terms together. Merging $3x + 5x^2$ into $8x^3$? Never — the classic forbidden combining like terms move. The variable parts differ, so not one step may merge. $3x + 5x^2$ is already simplest.

2. Treating addition as multiplication: $x + x$ is not $x^2$. $x \times x = x^2$ is multiplication; $x + x = 2x$ is addition. Combining like terms touches coefficients only, never exponents. The flip side too: $x^2 + x^2 = 2x^2$, not $x^4$.

3. Forgetting the hidden coefficient. In $7x - x$, the bare $x$ is $1x$. The coefficient arithmetic reads $7 - 1 = 6$, so the combining like terms answer is $6x$ — not $7x$.

4. Dropping the sign while rearranging. $10a - 3 + 4a$ regroups to $(10a + 4a) - 3 = 14a - 3$. Writing $14a + 3$ means the $-3$ moved but its minus sign stayed behind — in combining like terms, the sign always travels with its term.

5. Adding numbers to letter terms. In $3 + 4x$, the 3 and the $4x$ are unlike — no $7x$ here. Constants combine only with constants, a boundary every combining like terms step respects.