Algebra - adding and subtracting

Some students find the topic of algebra daunting when they meet it for the first time, but it needn’t be if they know how to distinguish between like and unlike terms for the purpose of adding and subtracting in algebra.

So…

What is algebra and how do we add and subtract algebraic terms?

Algebra is a topic in mathematics in which letters and other symbols are used to represent numbers. In this blog, letter symbols are used to explain how we add and subtract algebraic terms.

Key points to remember when adding and subtracting algebraic terms.

  • A term contains a number, a letter symbol or a product of both. Example, 7, x and 7x are three terms. 7 is a number term, x is a letter term and 7x is a product of both.
  • A letter term without a number (coefficient) appearing before it means one lot of that letter term. In the example above, the term x can be written as 1x.
  • An expression is a collection of terms with at least one operation. Example, 2x + 3 is an expression. In this expression, we have a letter term 2x, a number term 3 and an operation of adding.
  • We can simplify an expression by adding and/or subtracting like terms.
  • Like terms are letter symbols of the same type. In the first example above, x and 7x are like terms because they both contain the same letter symbol x. Similarly, a2, -5a2 and 3a2 are like terms because they all contain the letter symbol a2.
  • We cannot add or subtract, unlike terms.
  • Unlike terms have different letter symbols or contain a letter symbol and a number. Example, 2x + 3 cannot be simplified by adding the terms together because they are unlike terms – one is a letter term and the other is a number term. Similarly, a, a2, and a3 are all unlike terms because they have different powers – they can’t be added or subtracted.

Now…

How do you simplify an expression?

You can start by rewriting the expression by collecting like terms. You simply write letters of the same type next to each other, then add or subtract them.

Below are five examples of simplifying algebraic expressions.

Example (1)

Simplify                                  2a + 3b + a + 4b

Collecting like terms gives:    2a + a + 3b + 4b

Simplifying gives:                   3a + 7b

Remember:                              The letter a means one lot of a and can be written as 1a.

Example (2)

Simplify                                  7x + 4 – 2x + 5

Collecting like terms gives:    7x – 2x + 4 + 5

Simplifying gives:                   5x + 9

Example (3)

Simplify                                  12 – 2p -14 – p

Collecting like terms gives:    – 2p – p + 12 – 14

Simplifying gives:                   – 3p – 2

Remember:                              The term – p means one lot of – p and can be written as – 1p.

Example (4)

Simplify                                  5a2 – 7a + 2a2 – 3a

Collecting like terms gives:     5a2 + 2a2 – 7a – 3a

Simplifying gives:                    7a2 – 10a

Example (5)

Simplify                                  7ab + cd – 2ab + 3cd

Collecting like terms gives:     7ab – 2ab + cd + 3cd

Simplifying gives:                   5ab + 4cd

Remember:                              The term cd means one lot of cd and can be written as 1cd.

Did you know…

You can use your knowledge of adding and subtracting integers to add and subtract algebraic terms.

Let us consider the calculation of adding (- a) four times.

This can be written as (- a) + (- a) + (- a) + (- a)

The result will be – 4a

The brackets have been used to show the type of letter term being added – a negative letter term.

It is worth remembering that adding a negative term gives the same result as subtracting the term.

i.e. + (-) means –

So (- a) + (- a) + (- a) + (- a) can be written as – a – a – a – a = – 4a

Another way of writing the calculation is (- 1a) + (- 1a) + (- 1a) + (- 1a) = – 4a.

– a means one lot of negative a, that is why it can be written as -1a.

 So what happens when you subtract a negative algebraic term?

Let us consider the calculation (- 5a) – (- 2a).

In this calculation, we are subtracting – 2a from – 5a.

The result will be – 3a.

It is worth remembering that subtracting a negative term gives the same result as adding the term.

i.e. – (-) means +

So (- 5a) – (- 2a) can be written as – 5a + 2a

Giving the answer of – 3a.

You may find it beneficial to see our blog on adding and subtracting integers.

We have covered some key points here in algebra with a focus on adding and subtracting algebraic terms. Please like and share this blog if you have found it useful. It was written by Key Stage Tutors a one-to-one tuition website providing online tuition in several academic subjects.

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