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

10 Mars bars cost £2.50 or 250 pence, can be rewritten as: 10m = 250 (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)!!

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).

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)

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

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!!

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?

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

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!!!

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

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

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.

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

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

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

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

–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

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)

Combining these terms gives 9x2 – 9x – 18 (3x +3)(3x – 6) F gives 9x2 O gives – 18x I gives + 9x 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!!

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!!

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