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

Algebraic Language A Pronumeral is a letter that represents an unknown number. For example, X might represent the number of school days in a year. A Term usually contains products or divisions of pronumerals and numbers. The term 3x2 means 3 x x x x or 3 lots of x2. The Coefficient is the number by which a pronumeral or product of pronumerals is multiplied. 3 is the coefficient of x2 in the term 3x2 Like Terms have exactly the same letter make up, other than order. 6x2 y and 14x2 y are like terms. A Constant Term is a number by itself without a pronumeral. -7 is the constant term in the expression 4x2 – x – 7.

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

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

Formulas & Substitution Example 1: If the perimeter of a square is given by the formula P = 4x, find the perimeter if x = 5 cm Solution: P = 4x, P = 4 x 5 Perimeter = 20 cm Example 2: If a gardener works out his fee by the formula C = 10 + 20h where h is the number of hours he works, work out how much he charges for a job that takes 4 hours. Solution: C = 10 + 20h C = 10 + 20x 4 Charge = $90

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

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

Rules Curvy We usually don’t write a times sign eg 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 eg 5y not y5 Letters are written in alphabetical order eg 6abc rather than 6bca Instead of using a division sign ÷, we write the term as a fraction eg 6a ÷ y becomes

Simplifying Expressions Multiplying algebraic terms. Follow the rules and algebra is easy 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

Index Notation 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

Laws of Indices 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: or 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 example 6: 4( 2x - 3) - 5( x + 2) = 8x - 12 - 5x - 10 = 3x - 22 example 7: 4( 2x - 3) - 5( x - 2) = 8x - 12 - 5x + 10 = 3x - 2

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 ) NB: Always take out the highest common factor. x x x 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)

Patterns (s) (c) Formula: c = 2s + 1 Shape Number of cubes 1 2 3 4 n 30 100 49 (c) 3 +2 5 +2 7 +2 9 n 2 + 1 61 201 24 Formula: c = 2s + 1

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

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