ALGEBRA 1

Words and Symbols ‘Sum’ means add ‘Difference’ means subtract ‘Product’ means multiply Putting words into symbols Examples: the sum of p and q means p + q a number 5 times larger than b means 5b a number that exceeds r by w means r + w twice the sum of k and 4 means 2(k + 4) y minus 4 means y - 4 y less than 4 means 4 - y

The Language of Algebra Word Meaning Example Variable A letter or symbol used to represent a number or unknown value A = π r 2 has A and r as variables Algebraic Expression A statement using numerals, variables and operation signs 3a + 2b - c Equation An algebraic statement containing an “ = “ sign 2x + 5 = 8 This is all precious Inequation An algebraic statement containing an inequality sign, e.g <, ≤, >, ≥ 3x - 8 < 2

Terms The items in an algebraic expression separated by + and - signs 4x, 2y2 3xy, 7 Like Terms Two terms that have EXACTLY the same variables (unknowns) 4x and -7x 3x2 and x 2 NOT x and x2 Constant Term A term that is only a number -7, 54 Coefficient The number (including the sign) in front of the variable in a term 4 is the coefficient of 4x2, -7 is the coefficient of -7y

Simplifying Expressions Multiplying algebraic terms. Follow my rules and algebra is easy Multiply the coefficients together. Simplify the unknowns by writing the letters in alphabetical order The terms do not have to be ‘like’ to be multiplied examples: f x 4 = 4a x 2b = 4f 8 ab -2a x 3b x 4c = -24 abc 2 x a + b x 3 = 2a + 3b

Rules Curvy We usually don’t write a times sign e.g 5y not 5 x y, 5(2a + 6) The unknown x is best written as x rather than x Numbers are written in front of unknowns e.g. 5y not y5 Letters are written in alphabetical order Instead of using a division sign ÷, we write the term as a fraction eg 6a ÷ y becomes

A number owns the sign in front of it Like Terms ‘Like terms’ are terms which have the same letter or letters (and the same powers) in them. ie when you say them - they sound the same. A number owns the sign in front of it We can only add and subtract ‘like terms’ Examples: 5x + 7x = 12x 5a + 3b - 2a - 6b = 3a - 3b 10abc - 3cab = 7abc (or 7bca or 7cba etc) -4x2 - 2 + 3x + 5 - x + 7x2 = 1 3x2 + 2x + 3 Note: x means 1x

Like terms should sound the same Collecting Like Terms Like terms should sound the same Adding like terms is like adding hamburgers. e.g. + gives You’ve started with hamburgers, added some more and you end up with a lot of hamburgers 2x + 3x = 5x You started with x, added more x and end up with a lot of x, NOT x2

Substitution into formulae Input, x 2 Putting any number, x into the machine, it calculates 5x - 7. i.e. it multiplies x by 5 then subtracts 7 2 3 3 e.g. when x = 2 3 e.g Calculate 5a - 7 when a = 6 6 5 x - 7 = 30 - 7= 23 e.g. Calculate y2 - y + 7 when y = 4 Writing 4 where y occurs in the equation gives 42 - 4 + 7 = 19

Substitution If x = 4 and y = -2 and z = 3 - eg 1: x + y + z = 4 + - 2 = 4 - 2 + 3 eg 4: 2 x2 = 2 x 42 = 5 = 2 x 16 eg 2: xy (z - x) = 4 x -2 ( 3 - 4) = 32 x -1 = 4 x -2 x -1 eg 5: (2x)2 = (2 x 4)2 = 8 = 82 = 64 eg 3: y2 = ( -2)2 = 4

Patterns Take the common difference to the rule. Unit Number (n) Number of cubes (c) 1 2 3 4 n c = 30 100 49 3 +2 5 +2 7 +2 9 2 n + 1 61 201 24 Take the common difference to the rule. Multiply by the letter. Look for what you need to add or subtract to make it correct

Patterns - example 2 Unit Number (n) Matchsticks (m) 1 2 3 4 n m= 40 41 6 + 5 11 + 5 16 + 5 21 5 n + 1 201 8

Patterns - example 3 Unit Number (n) Number of dots (d) 1 2 6 3 10 4 14 n d= 18 102 + 4 + 4 + 4 4 n - 2 70 26

Indices in Algebra 24 24 means 2 x 2 x 2 x 2 = 16 a x a x a = a 3 index, power or exponent 24 base a x a x a = a 3 2 x a x a x a x b x b = 2 a 3 b 2 m x m - 5 x n x n = m 2 - 5 n 2 When multiplying terms with the same base we add the powers eg 1: y 3 x y 4 = y 7 28 eg 2: 2 3 x 2 5 = eg 3: 3 m 2 x 2 m 3 = 6 m 5

When dividing terms with the same base we subtract the powers. eg 1: p 8 ÷ p 2 = p 6 5 eg 2: 5 x 4 1 2 m eg 3: 3

Expanding Brackets Each term inside the bracket is multiplied by the term outside the bracket. Remember: means the terms are multiplied example 1: 4 ( x + 2) = + 4x 8 example 2: x ( x - 4) = - x 2 4x example 3: 3y ( y 2 + y - 3) = 3y 3 + 3y 2 - 9y example 4: -2 (m - 4) = -2m + 8 NB: -2 x -4 = +8 example 5: x (x - 5) + 2 (x + 3) = x 2 - 5x + 2x + 6 = x 2 - 3x + 6

Factorising This means writing an expression with brackets eg 1: 2x + 2y = 2 ( ) x + y eg 2: 3x + 12 = 3 ( x + 4 ) eg 3: 6x - 15 = 3 ( 2x - 5 ) eg 4: 4x2 + 8x = 4 x ( x + 2 ) x x x NB: Always take out the highest common factor. eg 5: 10d 2 - 5d = 5d ( 2d - 1) eg 6: 12a 3b 4c 2 - 20a 2b 3c 3 + 8a 4b 4c 5 aaabbbbcc aabbbccc aaaabbbbccccc = 4a 2b 3c 2 ( 3a b - 5 c + 2a 2 bc 3)