Simplifying Expressions

What does "simplify" mean?

To simplify an algebraic expression means to write it in the shortest, neatest form without changing its value. We do this mainly by collecting like terms — gathering together the parts that are the same and combining them into one.

Simplifying makes expressions easier to read, easier to substitute into, and easier to solve. It is one of the first skills you build on top of Algebra Basics.

Terms, coefficients and like terms

An expression is made of terms, the pieces separated by + or − signs. In the expression

3x + 5y − 2

the terms are 3x, 5y, and −2.

• The number in front of a variable is the coefficient (the 3 in 3x).
• A plain number with no variable is a constant (the −2).

Like terms have exactly the same variable part — the same letters raised to the same powers. The coefficients can differ.

• 3x and 8x are like terms (both just x).
• 4y and 4y are like terms.
• 3x and 3y are not like terms (different letters).
• 2x and 2x² are not like terms (different powers).

Only like terms can be combined.

The rule for combining like terms

To combine like terms, add or subtract the coefficients and keep the variable part exactly the same.

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

Think of it concretely: 3 apples plus 5 apples is 8 apples. The "x" is the apple. You never get x² from adding x's, just as adding apples never gives you apple-squared.

A term written as just x has a hidden coefficient of 1, and a term written as −x has a coefficient of −1. Remember this when you count.

Worked example 1 — one variable

Simplify 7x + 2x − 4x.

1. All three terms are like terms (all just x).
2. Work left to right with the coefficients: 7 + 2 − 4.
3. 7 + 2 = 9, then 9 − 4 = 5.
4. Keep the variable: 5x.

Worked example 2 — watch the signs

Simplify 9y − 2y + y.

Each term keeps the sign written in front of it.

1. Coefficients: +9, −2, and +1 (because y means 1y).
2. Combine: 9 − 2 + 1 = 8.
3. Result: 8y.

The most common slip here is forgetting that a lone y counts as 1. Always picture the invisible 1.

Worked example 3 — two different variables

Simplify 5a + 3b − 2a + 4b.

When there are two kinds of term, sort them into groups first.

1. Group the a terms: 5a − 2a.
2. Group the b terms: 3b + 4b.
3. Combine each group: 5a − 2a = 3a, and 3b + 4b = 7b.
4. Write them together: 3a + 7b.

You cannot combine 3a and 7b — they are different variables, so the simplest form keeps both.

Worked example 4 — variables and constants mixed

Simplify 6 + 4x − 9 + x.

1. Identify the groups: x terms are 4x and x; constants are 6 and −9.
2. Combine the x terms: 4x + x = 5x (remember x = 1x).
3. Combine the constants: 6 − 9 = −3.
4. Result: 5x − 3.

By convention we write the variable term first and the constant last, so 5x − 3 rather than −3 + 5x.

Worked example 5 — a longer one

Simplify 2x + 7 + 3y − 5 − x + y.

1. x terms: 2x − x = 1x = x.
2. y terms: 3y + y = 4y.
3. Constants: 7 − 5 = 2.
4. Put it all together: x + 4y + 2.

A neat strategy for long expressions is to lightly underline each type of term in a different way before you combine, so none get lost.

Activity — simplify each expression

1. 6x + 2x
2. 10a − 3a − a
3. 4p + 5q − 2p
4. 8 + 3m − 5 + 2m
5. 7x + 4y − 3x + 2y − 1
6. n + n + n

Answers: 1) 8x 2) 6a 3) 2p + 5q 4) 5m + 3 5) 4x + 6y − 1 6) 3n

Why this matters

Simplifying is the housekeeping of algebra. Before you can solve a linear equation, you usually have to collect like terms so each side is as simple as possible. A tidy expression is far less likely to lead to a careless mistake.

The whole skill rests on one idea: only like terms combine, and when they do, you only change the coefficient. Keep the signs attached to their terms, treat a lone variable as having a coefficient of 1, and group before you add. Do that and simplifying becomes almost automatic.