Simplifying Expressions

$4x+3-7x+8$

is an expression. It has no equals sign because it is not claiming that two things are equal.

Simplifying means rewriting an expression more compactly without changing its value. One way to do this is to collect like terms. In the example above, $4x$ and $-7x$ can be combined, as can the constant terms $3$ and $8$.

Reading an expression as separate terms

A term is one part of an expression. Read the operation sign immediately before a term as belonging to that term. Thus

$8y+4xy+3a^2-7a+54$

contains the five terms $8y$, $4xy$, $3a^2$, $-7a$ and $54$. The first term has an unwritten positive sign, so $8y$ could be read as $+8y$.

A term is usually a product of a number and a variable part. In $-5x^2y$, the number $-5$ is the coefficient and $x^2y$ is the variable part. A variable written without a number has coefficient $1$, so $x=1x$ and $-x=-1x$. A number without variables, such as $54$, is a constant term.

How to recognise like terms

Like terms have exactly the same variable part: the same variables raised to the same powers. Their coefficients may be different.

For example:

• $3x$ and $-8x$ are like terms;
• $5a^2b$ and $\frac12a^2b$ are like terms;
• $7$ and $-4$ are like terms because both are constants;
• $x$ and $x^2$ are not like terms;
• $y$ and $xy$ are not like terms;
• $a^2b$ and $ab^2$ are not like terms.

The order of multiplied variables does not matter, so $xy$ and $yx$ have the same variable part. However, the powers still matter.

Why coefficients can be collected

Consider $3x+5x$. This means three lots of $x$ plus five lots of $x$, making eight lots of $x$:

$3x+5x=(3+5)x=8x.$

Only the coefficients are added or subtracted; the shared variable part stays unchanged. This is an application of the distributive law in reverse.

By contrast, $3x+5y$ cannot be combined because the two terms do not represent lots of the same quantity. It is similar to being unable to describe three apples and five pears simply as eight apples or eight pears.

A reliable method

To simplify a sum or difference of terms:

1. Identify each term, keeping its sign attached.
2. Sort the terms into groups with matching variable parts; put all constants together.
3. Add or subtract the coefficients within each group.
4. Write the remaining unlike terms as one expression.
5. Check that no further like terms remain.

For example, simplify

$4x^2+7x-2x-3+2x^2+3.$

Group matching terms while preserving their signs:

$4x^2+2x^2+7x-2x-3+3.$

Now collect each group:

$6x^2+5x+0=6x^2+5x.$

The $x^2$ terms and $x$ terms remain separate because $x^2$ and $x$ are different variable parts. The constants cancel because $-3+3=0$.

Collecting terms with several variables

The same method works when terms contain more than one variable. For example,

$6ab-4a+3ab+9-2a-5$

can be grouped as

$(6ab+3ab)+(-4a-2a)+(9-5).$

Therefore,

$6ab-4a+3ab+9-2a-5=9ab-6a+4.$

The $ab$ terms cannot be combined with the $a$ terms: $ab$ contains both $a$ and $b$, whereas $a$ does not.

Negative and fractional coefficients

Coefficients do not have to be positive whole numbers. For instance,

$\frac12p+3p-\frac54p$

contains three like terms, so collect their coefficients:

$\left(\frac12+3-\frac54\right)p=\frac94p.$

When subtracting, pay particular attention to the signs. For example,

$2m-7m=-5m,$

because $2-7=-5$.

Why rearranging is allowed

In a sum, terms may be reordered to place like terms together, provided each term keeps its sign. For example,

$4x+11-8x+2+x^2$

may be rearranged as

$x^2+4x-8x+11+2.$

It then simplifies to

$x^2-4x+13.$

The value has not changed: the same signed terms have merely been reordered and combined.

Checking that the value is unchanged

A simplified expression must be equivalent to the original for every permitted value of its variables. As a quick check of

$4x+11-8x+2+x^2=x^2-4x+13,$

choose, for example, $x=2$. The original gives

$4(2)+11-8(2)+2+2^2=9,$

and the simplified expression gives

$2^2-4(2)+13=9.$

Matching values support the simplification, although the algebraic collection of coefficients is what establishes that the expressions are equivalent.