1. The use of letters to replace a word and/or an unknown value
Basic Algebra
The use of letters to replace a word and/or an unknown value

2. 10 Mars bars cost £2.50 or 250 pence, can be rewritten as:
10m = (pence)
This means EXACTLY the same thing as the sentence but can be easier to read, especially if the statement is a lot longer (next slide)!!

3. E.g. 6 eggs, 10 cakes, 5 apples and 4 pears cost a total of £4.25
Can be written as:
6e + 10c + 5a + 4p = 4.25
The letter stands for an object (eggs, cakes, etc.) and the letter has a value (or individual cost).

4. 10m = 250 From this we can work out the cost of m (one Mars bar)
If we look again at the first statement (now called an EQUATION because of the equal sign)
10m = 250
From this we can work out the cost of m (one Mars bar)

5. If 10m = 250
Then m (we do not need to put in the “1”, more on this later) is simply:
m = 250 ÷ 10 = 25
Since part of the original statement gave the cost in pence, then the answer must be given in the appropriate units (pence)!
m = 25p

6. As long as you remember that the letters in algebra stand for objects and that those objects have a value, then things should be a bit easier!!

7. Simplification of Terms
This is just getting the same ‘bits’ (or like terms) together…
Example: 5a + 3b – 2a + 6b
Remember “a” could be apples and “b” could be bananas. 5a means 5 × a but we drop the × sign as it could be confused with the letter “x”. Think if we didn’t drop the times sign how could we show 6x…..6×x?

8. Also we show a single item “a” as simply the lone letter…we do not put a “1” in front. There is no need. ‘a’ indicates one apple (or 1a) so using a “1” is simply a waste of pen!!!
Back to the example 5a + 3b – 2a +6b
Get the like terms (or like ‘bits’) together and remember that the sign joins the letter that follows!!!!
5a – 2a + 3b + 6b
Do the maths, remember about negative numbers!!
3a + 9b

9. If a letter has a power, such as x3, then treat it as a separate ‘object’ or animal from similar terms such as 6x or 2x2 for example, UNLESS we are required to multiply or divide the terms!!!

10. Now try these questions
2x+3y-x+2y
3c+5d-c+6d+4c
4p-3q+2p+5q
5r-3s-7s+2r-s
5a-7b-4a-2b+12b
5ef+7gh-9ef+2gh
x + 5y
6c + 11d
6p + 2q
7r - 11s
a + 3b
- 4ef + 9gh

11. Multiplying Expressions
Example:
2a × 4a becomes 8a2
Multiply the numbers then the letter!!
Just like 2 × 2 = 22
So a × a = a2
And so 2a x 4a must equal 8a2

12. If we multiply, for example, 3a2 × 2a3, we do the same things;
That is the numbers first (3 × 2) then the powers (a2 × a3). Just remember the rules on multiplying powers. Which is……….?
3a2 × 2a3 = 6a5
If you can’t remember, write out the expression in its simplest terms.

13. Get the same ‘bits’ together
3a2 × 2a3
3 × a × a × 2 × a × a × a
Get the same ‘bits’ together
3 × 2 × a × a × a × a × a
This ‘bit’ means a5
Multiplying out gives:
6a5

14. What about if there are different letters to be multiplied?
No problem; just do as the previous slide….
Example:
4a × 3b becomes 12 ab
4 × a × 3 × b
4 × 3 × a × b
12ab
Get the same ‘bits together, then do the maths

15. A bit about convention on writing expressions in algebra.
Numbers in front of the letters
Letters in alphabetic order
High powers first
‘Lone’ numbers last
Example: 3ax3 – 2tx2 + 5px -7

16. Slightly harder algebra
Expanding brackets. 2(3a + 4b)
REMEMBER everything inside is multiplied by outside the brackets.
2(3a + 4b)
This means… 2 × 3a × 4b
6a + 8b

17. –2a(3a – 2b) – 2a × 3a – 2a × – 2b –6a2 + 4ab
One thing you must remember is the sign of what is outside the bracket, in the previous example was not given, so it is assumed to be +
–2a(3a – 2b)
– 2a × 3a – 2a × – 2b
–6a2 + 4ab
Remember that the – sign between the 3a and the 2b relates to the 2b

18. Sometimes we need to multiply out a pair of brackets
Sometimes we need to multiply out a pair of brackets. Remember the phrase FOIL.
F First
O Outer
I Inner
L Last
(3x +3)(3x – 6)

19. Combining these terms gives 9x2 – 9x – 18 (3x +3)(3x – 6)
F gives 9x2
O gives – 18x
I gives x
L gives – 18
Giving 9x2 – 18x + 9x – 18
Combining these terms gives
9x2 – 9x – 18
(3x +3)(3x – 6)
This is a quadratic expression---more later!!

20. There are several other methods, such as the ‘smiley face’.
(3x + 3)(3x – 6)
Whichever method you use you will get the same answer. Keep with it if you are happy and always get the right answer!!

21. Now try these 2(a+3) 2a + 6 3(2p+3) 6p + 9 7(4x-3) 28x – 21
9(3b+4c-2d)
Expand AND simplify
4(4a+5)-4(5a+2b)
(a+1)(a+2)
(3e-5)(2e-1)
(3p-1)(3p+1)
2a + 6
6p + 9
28x – 21
27b + 36c – 18d
– 4a – 8b +20
a2 +3a +2
6e2 – 13e + 5
9p2 – 1