C2.2; E2.2 - Collecting like terms

Terms and coefficients

Why can $3x+2x$ become $5x$, while $3x+2$ usually cannot become a single term? Think of $x$ as the number of counters in one packet. Three packets plus two packets make five packets. Three packets plus two loose counters still need both descriptions.

The letter represents a number of counters, rather than the packet itself. The packet picture helps you count equal quantities; the algebra also works with negative or fractional values of $x$, where a literal packet picture is less useful.

Keep each sign with its term. In $5a-b+9$, the term $-b$ means $(-1)b$, so its coefficient is $-1$. In $a+2$, the coefficient of $a$ is $1$, even though the $1$ is not written.

An addition or subtraction outside brackets separates terms. Multiplication does not: $4xy$ is one term containing the factors $4$, $x$ and $y$. In $3(x+2)$, the whole expression is a product; $x+2$ is kept together inside brackets. You will learn to remove such brackets in the next lesson.

Compare $6x^2$ with $(6x)^2$. The first has coefficient $6$; squaring the whole bracket in the second gives $36x^2$.

Collecting like terms

To simplify an expression, write it in an equivalent, simpler form. Equivalent expressions have the same value for every allowed choice of their variables. Collecting like terms counts copies of the same quantity.

For instance, $3x+2x=(3+2)x=5x$. This works because both terms contain the same multiplier $x$. You add the coefficients and retain that letter part. For subtraction, add the signed coefficients: $3x-5x=(3-5)x=-2x$.

Change only the final plus sign to a minus sign. The $a$-terms stay the same; the $b$-coefficient must change.

An expression is fully simplified by collecting when each different letter part appears at most once. A zero coefficient removes a term: $5a-5a=0a=0$. Also combine constants; for example, $2x+7+x-4=3x+3$.

A numerical check can expose a wrong simplification. To justify a correct one generally, identify and combine the matching letter parts.

Powers and mixed-variable expressions

Matching the letters is not enough: their powers must match too. Since $x^2=x\times x$, it is a different letter part from $x$. For $x=3$, one copy of $x^2$ is $9$, while one copy of $x$ is $3$.

The order of multiplication does not matter: $xy=yx$. Thus $3xy$ and $5yx$ are like terms and combine to $8xy$. But $x^2y$ and $xy^2$ are different. One contains two factors of $x$; the other contains two factors of $y$.

Before collecting, write a consistent letter order and inspect the whole letter part. Changing a coefficient counts copies; it does not change a power.

Now reverse the letters in one product. This changes its appearance, but does not change its value or its group.

Write higher powers first when this makes an answer easier to read, but order alone does not change the value. For example, $3+2x^2-x$ and $2x^2-x+3$ are equivalent.

This answers the opening question: only terms describing copies of the same letter part can be collected.

Quick Recap

• Keep a term's sign with it; the coefficient of $-x$ is $-1$.
• Like terms have the same letters and the same powers.
• Add their coefficients and keep their letter part unchanged.
• Products such as $ab$ and $ba$ match, but $a^2$, $a$ and $ab$ are different.

$3x+2x=5x$ counts five copies of the same quantity. In $3x+2$, the two terms describe different quantities, so both remain.