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Chapter 2: Algebra & Equations
Exercise 2A – Operations with Pronumerals
Like terms are terms that contain the same pronumerals and can be collected in order to
simplify an algebraic expression.
Class Worked Example
Example 1: Simplify 4j – 5c + c + 3j
a) Write the expression
4j – 5c +c + 3j
b) Identify like terms and group them together
4j + 3j – 5c + c
c) Simplify by collecting like terms
7j – 4c
When multiplying and dividing algebraic terms, it is not necessary to have like terms.
Example 2: Simplify 5m x -6p
a) Write expression
5m x -6p
b) Rearrange, writing coefficients first
5 x -6 x m x p
c) Multiply coefficients and pronumerals separately
-30mp
Example 3: Simplify a2b/6ab2
a) Write expression
a2b/6ab2
b) Cancel a from numerator and denominator
a2b/6ab2
ab/6b2
c) Cancel b from numerator and denominator
ab/6b2
a/6b
Exercise 2B – Substituting in Expressions
When the numerical values of pronumerals are known, we can substitute them into an
algebraic expression and evaluate it.
It is sometimes useful to place any substituted values in brackets when evaluating an
expression.
When dealing with numbers and pronumerals, particular rules must be obeyed:
1. The Communitive Law holds true for addition and multiplication as x + y = y+ x and x
xy=yxx
2. The Associative Law holds true for addition and multiplication as (x + y) + z = x + (y
+z) and (x x y) x z = x x (y x z)
3. The Identity Law states that in general x + 0 = x and x x 1 =x
4. The Inverse Law states that in general x + -x = 0 and x x (1/x) = 1
5. The Closure Law states that, when an operation is performed on an element, or
elements of a set
Exercise 2C - Expanding
Expanding brackets is an algebraic expression is achieved by multiplying the term outside
the brackets by each of the terms inside. This is called the Distributive Law.
Distributive Law = a(b+c) = ab + ac
Class Worked Example
Example 1: Expand 7(m-4)
a) Write expression
7(m-4)
b) Multiply each term inside the bracket by the term outside
7xm–7x4
7m – 28
Example 2: Expand and Simplify 6(m-4r) – 2(2m +7r)
a) Write expression
6(m-4r) – 2(2m +7r)
b) Multiply each term inside the bracket by the term outside
6 x m – 6 x 4r – 2 x 2m + (-2) x 7r
6m – 24r – 4m - 14r
c) Simplify by collecting like terms
2m – 38r
Complete 1-6, 7
Exercise 2D – Factorising using Common Factors
Factorising is the opposite process to expanding.
Example 1: Factorise the following 6a-15
a) Write the expression
6a – 15
b) Find the highest common factor (HCF) of terms
HCF = 3
c) Write each term in the expression as a product of 2 factors, one being the HCF
6a – 15
3 x 2a – 3 x 5
d) Place HCF outside a pair of brackets with remaining terms inside
3(2a – 5)
Class Worked Example
Example 2: Factorise 20p6 + 15p4
a) Write the expression
20p6 + 15p4
b) Find HCF
HCF = 5 and p4 = 5p4
c) Write each term as a product of two factors, 1 being the HCF
5p4 x 4p2 + 5p4 x 3
d) Place HCF outside brackets and remaining terms inside
5p4(4p2 + 3)
Complete questions 1-5
Exercise 2E – Adding & Subtracting Algebraic Fractions
The methods for dealing with algebraic fractions are the same as those used for numerical
fractions. To add or subtract algebraic fractions we perform these steps:
1. Find lowest common denominator (LCD) by finding the lowest common multiple
(LCM) of the denominators.
2. Rewrite each fraction as an equivalent fraction with this common denominator.
3. Express as a single fraction
4. Simplify numerator
Example 1: Simplify the following ((x+1)/5) + ((x+4)/3)
a) Write the expression
((x+1)/5) + ((x+4)/3)
b) Rewrite each fraction as an equivalent fraction using the LCD
((x + 1)/5) + ((x + 4)/3)
((x + 1)/5) x 3 + ((x +4)/3) x 5
(3(x + 1)/15) + (5(x +4)/15)
c) Express as a single fraction
(3(x + 1) + 5(x + 4)) / 15
d) Simplify numerator by expanding brackets and collecting like terms
(3x + 3 + 5x + 20) / 15
(8x + 23) / 15
Complete questions 1-3
Exercise 2F – Multiplying & Dividing Algebraic Fractions
The rules for multiplication and division are the same as for numerical fractions. When
multiplying algebraic fractions, multiply the numerators, then multiply the denominators
and cancel any common factors if possible.
When dividing algebraic fractions, change the division sign to a multiplication and write the
following fraction as its reciprocal.
Example 1: Simplify 2x / ((x+1)(2x-3)) x ((x+1)/x)
a) Write the expression
2x / ((x+1)(2X-3) x ((x+1)/x)
b) Multiply numerators then the denominators
2x(x+1) / (x(x+1)(2x-3))
c) Check for common factors and cancel
Cancel out both (x +1)’s
2x / x(2x – 3)
Cancel out x’s
2/(2x – 3)
Complete questions 1-4
Exercise 2G – Solving Basic Equations
Equations are algebraic sentences that can be solved to give a numerical solution.
Remember: to solve any equation we need to undo all the operations that have been
performed on the pronumeral.
Example1: Solve 5y – 6 = 79
a) Write expression
5y – 6 =79
b) Add six to both sides
5y – 6 + 6 = 79 + 6
5y = 79
c) Divide both sides by 5
5y/5 = 79/5
y = 17
Equations where Pronumerals are on both Sides
We can also solve equations where the pronumerals appear on both sides of the equation.
It is the aim to aid or subtract one of the pronumeral terms so that it is eliminated from one
side of the equation.
Example 2: Solve 14 –4d = 27 – d
a) Write equation
14 – 4d = 27 – d
b) Create a single pronumeral term by adding 4d to both sides
14 – 4d + 4d = 27 – d + 4d
14 = 27 – 3d
c) Subtract 27 from both sides of the equation
14 – 27 = 27 –27 – 3d
-13 = 3d
d) Divide both sides by 3
-13/3 =3d/3
d = -13/3
e) Express improper fraction as a mixed number
D = -41/3
Complete questions 1-10, 11-14
Exercise 2H – Solving More Complex Equations
Equations with Multiple Brackets
Many equations need to be simplified by expanding brackets and collecting like terms
before they are solved.
Class Worked Example
Example 1: Solve 6(x+1) – 4(x-2) = 0
a) Write the expression
6(x+1) – 4(x-2) = 0
b) Expand all brackets
6 x x + 6 x 1 – 4 x x –4 x –2 = 0
6x + 6 – 4x +8 = 0
c) Collect like terms
2x + 14 = 0
d) Subtract 14 from both sides
2x + 14 –14 = 0 –14
2x = -14
e) Divide both sides by 2
2x / 2 = -14/2
x = -7
Equations with Algebraic Fractions
To solve equations with algebraic fractions, write every term in the equations as a
fraction with the same common denominator. Every term can then be multiplied by this
common denominator.
Example 2: Solve (x/2) – (3x/5) = (1/4)
a) Write expression
(x/2) – (3x/5) = (1/4)
b) Write each terms as an equivalent fraction with a denominator by 2
((x/2)x (10/10)) – ((3x/5) x (4/4)) = ((1/4)x (5/5))
(10x/20) – (12x/20) = (5/20)
c) Multiply each term by 20 to remove denominator
(10x/20) – (12x/20) = (5/20)
10x/20 x 20 – 12x/20 x 20 = 5/20 x 20
10x – 12x = 5
d) Simplify left hand side of equations by collecting like terms
10x – 12x = 5
-2x = 5
e) Divide both sides by –2
-2x/-2 = 5/-2
x = -5/2
f) Express improper fraction as a mixed number
X = -2 ½
Complete questions 1-3, 4
Exercise 2I – Solving Inequations
Inequations involved the inequality signs <, >, <, or >, in most cases an inequation can be
solved as if the inequality sign was an equals sign.
When multiplying or dividing by a negative number the inequality sign needs to be
reversed.
Class Worked Example
Example 1: Solve –3n/7 < 6
a) Write equation
-3n/7 < 6
b) Multiply both sides by 7
-3n/7 x 7 < 6 x 7
-3n < 42
c) Divide both sides by –3 and reverse sign as dividing by –ve
-3n/-3 < 42/-3
n > -14
The solution to an inequation can be graphed on a line. This is done by placing a circle
above the number which solves the matching equation together with an arrow in the
direction of the inequality.
< or > = hollow or open circle =
< or > = filled or closed circle =
Class Worked Example
Example 2 – Solve and sketch 7h + 4 > 5h-7
a) Write inequation
7h + 4 > 5h – 7
b) Subtract 5h on both sides
7h – 5h + 4 > 5h – 5h – 7
2h + 4 > -7
c) Subtract 4 from both sides
2h + 4 – 4 > -7 – 4
2h > -11
d) Divide both sides by 2
2h/2 > -11/2
h > -11/2 or –5 ½
e) Sketch – with a
circle as >
Complete questions 4-6