Simplifying Expressions and Collecting Like Terms

To simplify an expression means to write it in a shorter way that has the same value. You do this by collecting like terms: adding or subtracting the terms that are the same kind of thing. For example, 3t + 2t = 5t, just as 3 tins plus 2 tins make 5 tins.

In the exam you are usually asked to simplify an expression such as 4k + 3 + 2k, for 1 or 2 marks, often near the start of a paper. You also collect like terms when you write an expression for a perimeter and when you solve equations.

A term is one part of an expression. Terms are separated by + and - signs. In 5x + 3 - 2y, the terms are 5x, +3 and -2y

The coefficient is the number in front of a letter. In 5x the coefficient is 5. A letter on its own has coefficient 1, so x means 1x and m + m + m + m = 4m. Numbers with no letter are collected together too.

Like terms have exactly the same letters, with the same powers. 3m and 8m are like terms. 3m and 3n are not, and nor are m and m^2. To collect like terms, add or subtract their coefficients and keep the letter part the same.

The sign in front of a term belongs to that term. In 5b + 2 - 3b - 6, the terms are +5b, +2, -3b and -6. You can move terms around to put like terms together, but always take the sign with them.

A coefficient can be negative: 3b - 5b = -2b. If like terms cancel, as in 4b - 4b = 0, leave them out of the answer.

x^2 means x \times x, which is a different kind of term from x. So x^2 and x are not like terms, and 4x^2 + 3x cannot be simplified. Collect the x^2 terms together and the x terms together.

Adding and multiplying give different answers: x + x = 2x, but x \times x = x^2

The perimeter of a shape is the total distance around its edge. When the sides are given as expressions, add all the sides and then collect like terms.

Sides with the same marks on them are equal in length, so each one must be counted. Brackets round a length, such as (2x + 3) cm, just keep that side's expression together.

• Turning repeated adding into a power. k + k + k is 3k, not k^3, which means k \times k \times k.
• Losing a minus sign. Keep each sign with the term after it, so 3y - 7y gives a negative answer.
• Forgetting the 1 in front of a letter on its own. x counts as 1x and -x counts as -1x when you collect.
• Joining unlike terms into one term. 8c + 6d is already fully simplified, so writing it as 14cd loses the mark.
• Changing the power when collecting. Add the coefficients, keep the power the same, and never collect an x^2 term with an x term.
• Leaving two signs side by side. Adding a negative term is a subtraction, so write 9 + -2b as 9 - 2b in your final answer.