Terms in an equation: what is a term?

An equation is split up into terms, which are separated by plus and minus. Knowing where one term stops and the next begins is what lets you simplify an equation without breaking it. The usual hurdle is the minus sign: it belongs to the term after it, and it has to travel with it.

When do I use this?

Every time you simplify or solve an equation. Before you can collect like terms, take a term away on both sides, or divide an equation by 2, you have to be able to see the terms. The same idea runs through simplifying expressions in general.

What counts as a term

Take this equation:

$$2x + 5 \times 3 - 4x = x + 53 \times x \times 2 - 45$$

If we draw a ring around each term, we get six rings: around $2x$, around $5 \times 3$, around $4x$, around $x$, around $53 \times x \times 2$ and around $45$. The plus and minus signs sit between the rings, not inside them. So here we see six terms, and the terms are split up by plus and minus.

Notice that a multiplication sign does not split a term. $5 \times 3$ is one term, and $53 \times x \times 2$ is one term, even though there are several things multiplied together inside it. Only plus and minus split.

The sign belongs to the term

Even though the rings sit around the numbers and not around the signs, it's still important to remember that the number and its sign belong together. The third term above is $-4x$, and it keeps its minus wherever it goes.

The same thing catches people out when a term is written the other way round. If it says

$$-x \times 4$$

that is minus $x$ times 4, so we end up with $-4x$, and not just $4x$. That's important to remember.

Multiplying or dividing an equation: all the terms at once

When we multiply or divide on both sides of an equation, we really do it through all the terms. But we can also just use brackets. If we wanted to divide this equation by 2, we could use brackets or do it for all the terms:

$$\frac{2x}{2} + \frac{5 \times 3}{2} - \frac{4x}{2} = \frac{x}{2} + \frac{53 \times x \times 2}{2} - \frac{45}{2}$$

or

$$\frac{(2x + 5 \times 3 - 4x)}{2} = \frac{(x + 53 \times x \times 2 - 45)}{2}$$

Or if we wanted to multiply by 2:

$$2x \times 2 + 5 \times 3 \times 2 - 4x \times 2 = x \times 2 + 53 \times x \times 2 \times 2 - 45 \times 2$$

$$(2x + 5 \times 3 - 4x) \times 2 = (x + 53 \times x \times 2 - 45) \times 2$$

Often we use brackets because it looks tidier, but we still have to multiply the 2 into the bracket, and then we actually end up with the same thing. How that works is in the guide on expanding a single bracket.

It's just as correct to use brackets as to multiply or divide through all the terms, but learning about the terms in an equation has its advantages. It gives a clearer picture of doing the same to all the terms at once, which is really also what we do with the brackets. It's just a bit more hidden.

Why terms matter for solving

The whole idea of solving equations is to find the opposite of what you want to get rid of, so you can get the unknown value on its own. And what we get rid of is always a whole term. We take $+12$ away, we take $2x$ away, we divide every term by 2. We never take away half a term. So once you can see the terms, you can see exactly which moves are allowed. The guides on solving linear equations and equations with the unknown on both sides put this to work.

Common mistakes

• "$-4x$ and $4x$ are the same term." No. The number and its sign belong together. The ring goes around $4x$, but the minus in front of it comes along.
• "I divided the equation by 2, so I divided the first term by 2." When we divide on both sides, we divide all the terms, on both sides. Or we put each side in a bracket and divide the whole bracket.
• "$5 \times 3$ is two terms." No. A multiplication sign does not split a term. Only plus and minus do, so $5 \times 3$ is one term.

Related

• What is an equation?
• How to solve linear equations
• Equations with the unknown on both sides
• Simplifying algebraic expressions