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 = 7If n = 3 , the expression = 9

• AlgebraAlgebra applies quantitative concepts to unknown quantities represented by symbols.

• A termterm is a part of an expression that is connected to another term by a plus or minus sign.

• A constantconstant is a term whose value does not change.

• A variablevariable is a term that represents a quantity that may have different values.

• An expressionexpression is a combination of constants and variables using arithmetic operations.

Definitions

Definitions

• A coefficientcoefficient is a factor by which the rest of a term is multiplied.

• The degreedegree of expression is the highest exponent of any variable in the expression.

• An equationequation is a statement that two expressions are equal.

Algebraic Expressions

• Terms are added together

432 2 xx

3 Terms

Algebraic Expressions

• Factors are multiplied

Algebraic ExpressionsHow many factors in each

term?

x 432 2 x

1 Factor2 Factors3 Factors

Coefficients

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

Algebraic ExpressionsWhat is the coefficient of x?

x 432 2 x

Coefficient

Algebraic ExpressionsWhat is the coefficient of

x2?

x 432 2 x

Coefficient

Like Terms

• Like terms have the same power of the same variable(s)

x2 2x5 2 x 2

x2But not y3 xy2, ,

, ,

Combining Like Terms

• Distributive Law– ab + ac = a(b+c) = (b+c)a

x2 2x5 2+ x7 2=

x2 2x5 2+ x(5+2) 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:

xx-x

x-x

xx

235

combined bemay 3and5

termslikeare3and5

Algebraic rules

Rule:• When an expression is contained in brackets,

each term within the brackets is multiplied by any coefficient outside the brackets.

Example:

286124232

bracketstheremoveto

1432

:expressiontheConsider

yxyx

yx

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:

26

2436

1221231223

1223

2

2

xx

xxx

xxxxx

xandx

sexpressiontwotheofproductThe

Minus Signs

• 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

3x – 2 = 3x + ( 2)3x – 2 = ( 2) + 3x

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

Algebraic rules

Solving linear equations

Solve 9x - 27 = 4x + 3 for x1. 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 + 2WIf 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 hoursW = 5 40 + 35= 200 + 35= £235

Substitution

• Find the value of 4y - 1 when• y = 1/40• y = 0.5 1• Find F = 5(v + 6) when v = 975

Rearranging formulae

• Sometimes it is easier to use a formula if you rearrange it first

• y = 2x + 8• Make x the subject of the formulaSubtract 8 from both sidesy 8 = 2xDivide both sides by 2´2y - 4 = x

Rearranging formulae

• A = 3r2

Make r the subject of the formulaDivide both sides by 3A/3 = r2

Take the square root of both sides 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 55(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)?= -2 x - (-2) 3y= -2x + 6y

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)