Algebra

Algebra Algebra is a way of representing numbers with letters, rather than using numbers themselves This means you can generalise calculations

Variables Letters represent an unknown or generic real number Sometimes with restrictions, such as “a positive number”

Variables Often a letter from the end of the alphabet: x, y, z Or that stands for a quantity: d for distance, t for time, etc.

Algebraic Expressions Expressions do not include an equal sign An algebraic expression equals a number (depending on the variables)

Algebraic Expressions Example: 2n + 3 If n = 1 , the expression = 5 If n = 2 , the expression = 7 If n = 3 , the expression = 9

Definitions Algebra applies quantitative concepts to unknown quantities represented by symbols. A term is a part of an expression that is connected to another term by a plus or minus sign. A constant is a term whose value does not change. A variable is a term that represents a quantity that may have different values. An expression is a combination of constants and variables using arithmetic operations.

Definitions A coefficient is a factor by which the rest of a term is multiplied. The degree of expression is the highest exponent of any variable in the expression. An equation is a statement that two expressions are equal.

Algebraic Expressions Terms are added together 4 3 2 + - x 3 Terms

Algebraic Expressions Factors are multiplied

Algebraic Expressions How many factors in each term? 4 3 2 + - 3 Factors 2 Factors 1 Factor

Coefficients Coefficients are constant factors that multiply a variable or powers of a variable

Algebraic Expressions What is the coefficient of x? 4 3 2 + - Coefficient

Algebraic Expressions What is the coefficient of x2? 4 3 2 + - Coefficient

x 5 x 2 x , , x 2 , y 3 , xy 2 Like Terms 2 2 But not Like terms have the same power of the same variable(s) x 5 2 x 2 x 2 , , x 2 , y 3 , xy 2 But not

x 2 5 + x (5+2) = x 2 5 + x 7 = Combining Like Terms 2 2 Distributive Law ab + ac = a(b+c) = (b+c)a x 2 5 + x (5+2) 2 = x 2 5 + x 7 2 =

Algebraic rules Rule: If an expression contains like terms, these terms may be combined into a single term. Like terms are terms that differ only in their numerical coefficient. Constants may also be combined into a single constant. Example:

Algebraic rules Rule: When an expression is contained in brackets, each term within the brackets is multiplied by any coefficient outside the brackets. Example:

Algebraic rules Rule: To multiply one expression by another, multiply each term of one expression by each term of the other expression. The resulting expression is said to be the product of the two expressions. Example:

Minus Signs 3x - (2 - x) = 3x – 2 - -x = 3x - 2 + x = 4x - 2 Subtraction is Adding the Opposite A minus in front of parentheses switches the sign of all terms 3x - (2 - x) = 3x – 2 - -x = 3x - 2 + x = 4x - 2

Subtraction Adding the Opposite

Algebraic rules Rule: Any term may be transposed from one side of an equation to the other. When the transposition is made, the operator of the term must change from its original. ‘+’ becomes a ‘-’ and ‘-’ becomes a ‘+’. Example: 15x - 20 = 12 - 4x 15x - 20 + 4x = 12 15x + 4x = 12 + 20

Solving linear equations Solve 9x - 27 = 4x + 3 for x 1. Place like terms of the variable on the left side of the equation and the constant terms on the right side. 9x - 4x = 3 + 27 2. Collect like terms and constant terms. 5x = 30 3. Divide both sides of the equation by the coefficient of the variable (in this case 5). x = 6

Algebra - substitution or evaluation Given an algebraic equation, you can substitute real values for the representative values Perimeter of a rectangle is P = 2L + 2W If L = 3 and W = 5 then: P = 2 ´ 3 + 2 ´ 5 = 6 + 10 = 16

Substitution A joiner earns £W for working H hours Her boss uses the formula W = 5H + 35 to calculate her wage. Find her wage if she works for 40 hours W = 5 ´ 40 + 35 = 200 + 35 = £235

Substitution Find the value of 4y - 1 when y = 1/4 y = 0.5 1 y = 0.5 1 Find F = 5(v + 6) when v = 9 75

Rearranging formulae Sometimes it is easier to use a formula if you rearrange it first y = 2x + 8 Make x the subject of the formula Subtract 8 from both sides y - 8 = 2x Divide both sides by 2 ´2y - 4 = x

Rearranging formulae A = 3r2 Make r the subject of the formula Divide both sides by 3 A/3 = r2 Take the square root of both sides Ö A/3 = r

Brackets The milkman’s order is 3 loaves of bread, 4 pints of milk and 1 doz. Eggs per week Suppose the cost of bread is b, the cost of milk is m and a dozen eggs is e. Work out the cost after 5 weeks = 5(3b + 4m + e)

Brackets = 5(3b + 4m + e) to ‘remove’ brackets, each term must be multiplied by 5 5(3b + 4m + e) = 15b + 20m + 5e If the number outside the bracket is a negative, take care: the rules for multiplication of directed numbers must be applied

Brackets 4(3x - 2y) 4 ´ 3x - 4 ´ 2y = 12x - 8y What about -2(x-3y)?

Factorising The opposite of multiplying out brackets Need to find the common factors Very important - it enables you to simplify expressions and hence make it easier to solve them A factor is a number which will divide exactly into a given number.

Factorising 2x + 6y 2 is a factor of each term (part) of the expression and therefore of the whole number. 2x + 6y = 2 ´ x + 2 ´ 3 ´ y = 2 ´ (x +3 ´ y) = 2(x + 3y)

Factorising 6p + 3q + 9r 3 is the common factor = 3(2p + q + 3r) x2 + xy + 6x x is the common factor for each term x(x + y + 6)