For example, we can use y to represent any number.
It is important to understand that y means 1y even though we often write 1y as y.
2y means 2 × y, 3y means 3 × y, and 4y means 4 × y.
The digit is always written in front of the letter.
Algebraic Expressions
4p + 9
This is an algebraic expression containing 2 terms.
The term 4p contains a p but the term 9 doesn't.
We say 9 is a constant because its value is fixed and cannot change.
But the value of 4p is unknown because it depends on the value of p.
Simplifying Algebraic Expressions
Very often, we are asked to simplify an Algebraic Expression. All you have to do is put all the terms containing similar letters together. Constants (terms without letters) are also collected into a single term.
Example 1:
Simplify 20q + 11 + 9q.
Solution:
20q + 11 + 9q
= 20q + 9q + 11
= 29q + 11
Example 2:
Simplify 18r – 5 – 8r – 4.
Solution:
18r – 5 – 8r – 4
= 18r – 8r – 5 – 4
= 10r – 9
Example 3:
Simplify 30s – 21 + 15s + 4 – 20s.
Solution:
30s – 21 + 15s + 4 – 20s
= 30s + 15s – 20s – 21 + 4
= 25s – 17
Example 4:
Simplify 12t + 3 – 16t – 20 + 10t + 30.
Solution:
12t + 3 – 16t – 20 + 10t + 30
= 12t – 16t + 10t + 3 – 20 + 30
= 6t + 13
Note 1: A term without a + or – sign in front of it is assumed to be positive.
Note 2: In primary school, we usually avoid negative numbers as pupils have not been taught on this topic. But you should be familiar with them already because we use them when measuring temperatures, such as –10 degree Celsius.
Simplifying Algebraic Expressions involving multiplication
The term 4a means 4 times a, so: 4a = a + a + a + a = 4 × a
Therefore, 5 × 4a = 5 × 4 × a = 20 × a = 20a
Also, 11b × 6 = 11 × b × 6 = 66b
And, 20c × 10 + 1 = 20 × c × 10 + 1 = 200c + 1
But it is not necessary to write out everything the way I have done. I did it to show you what it means. Usually, we simplify such terms in one step:
Example 5:
Simplify 8 × 10d.
Solution:
8 × 10d
= 80d
Example 6:
Simplify 3 × 12e – 2e.
Solution:
3 × 12e – 2e
= 36e – 2e
= 34e
Note 3: BODMAS rules still apply when we are doing Algebra.
Note 4: We'll only be dealing with letters multiplied by numbers here. (I won't be dealing with cases where letters are multiplied by other letters because that's not in the primary school syllabus.)
Example 7:
Simplify 40f × 2 – 11 – 8f – 10.
Solution:
40f × 2 – 11 – 8f – 10
= 80f – 8f – 11 – 10
= 72f – 21
Example 8:
Simplify 12g – 30 – 15g × 2 + 5 + 50g – 20.
Solution:
12g – 30 – 15g × 2 + 5 + 50g – 20
= 12g – 30g + 50g – 30 + 5 – 20
= 32g – 45
Simplifying Algebraic Expressions involving division
In algebra, we usually express division in terms of a fraction.
Thus, h ÷ 2 is written as h⁄2
The 4 operation rules for algebra are the same as for whole numbers.
But we must be clear whether we are dealing with the numerators or denominators.
For example, in h⁄2, h is the numerator and 2 is the denominator.
Example 9:
Simplify 10j ÷ 6 – 7.
Solution:
10j ÷ 6 – 7
= 10j⁄6 – 7
= 5j⁄3 – 7
Example 10:
Simplify 5k ÷ 25 + 12.
Solution:
5k ÷ 25 + 12
= 5k⁄25 + 12
= k⁄5 + 12
Example 11:
Simplify 18m – 11 – 30m ÷ 6 + 7.
Solution:
18m – 11 – 30m ÷ 6 + 7
= 18m – 30m⁄6 – 11 + 7
= 18m – 5m – 4
= 13m – 4
Example 12:
Simplify 10m ÷ 15 – 24 + 36m ÷ 9 + 7.
Solution:
10m ÷ 15 – 24 + 36m ÷ 9 + 7
= 10m⁄15 + 36m⁄9 – 24 + 7
= 2m⁄3 + 4m – 17
= 14m⁄3 – 17
Simplifying Algebraic Expressions involving multiplication and division
When we multiply a fraction which has a letter, the numbers are dealt with as usual, but the letters are dealt with separately.
Example 13:
Simplify 4 × 6n ÷ 5 – 20
Solution:
4 × 6n ÷ 5 – 20
= 24n ÷ 5 – 20
= 24n⁄5 – 20
Example 14:
Simplify 11 + 9 × 2p ÷ 3 – 7
Solution:
11 + 6 × 4p ÷ 3 – 7
= 6 × 4p ÷ 3 + 11 – 7
= 2×4p⁄1 + 4
= 8p + 4
Note 5: The Left to Right Rule still applies. But in this case, we know that 3 is a denominator, so we can reduce 6 and 3 to 2 and 1.
Simplifying Algebraic Expressions with Brackets or Parentheses
When brackets or Parentheses are used to enclose an expression, it means everything inside the brackets is treated as a group. When a number is written just outside the brackets, it means that everything enclosed by the brackets is multiplied by the number.
Example 15:
Simplify 5(6q – 8) – 10.
5(6q – 8) – 10
= 5 × 6q – 5 × 8 – 10
= 30q – 40 – 10
= 30q – 50
Example 16:
Simplify 15r – 3(4r – 1) .
15r – 3(4r – 1)
= 15r – 3 × 4r + 3 × 1
= 15r – 12r + 3
= 3r + 3
Note 6: When we multiply a negative number by another negative number, we get a positive number. That's why –3 × –1 becomes + 3 × 1. The two negative signs sort of cancel each other out.
Example 17:
Simplify 55 – 8(10s – 5) – 3 + (2s – 7).
55 – 8(10s – 5) – 3 + (2s – 7)
= 55 – 8 × 10s – 8 × 5 – 3 + 2s – 7
= 55 – 80s – 40 – 3 + 2s – 7
= 55 – 40 – 3 – 7 – 80s + 2s
= 5 – 78s
Note 7: Usually, we put constants at the end. But in this case, the term with r is negative, so we put the constant in front. It is not wrong to put a negative term in front. Just a preference.
Example 18:
Simplify 3(6t⁄9 – 5) + 5
3(6t⁄9 – 12) + 5
= 3×6t⁄9 – 3 × 12 + 5
= 6t⁄3 – 36 + 5
= 2t – 31
