Download Algebraic Manipulation – AM1

Intro – From the Syllabus
Strand: Algebra and Modelling
Algebra involves the use of symbols to represent numbers or quantities and to express relationships. A
mathematical model is a mathematical representation of a situation. All applications of mathematics are based on
mathematical models. The study of Algebra and Modelling is important in developing students’ reasoning skills
and their ability to represent and solve problems.
In the Algebra and Modelling Strand in the Preliminary Mathematics General course, students apply algebraic skills
and techniques to interpret and use simple mathematical models of real situations.
Outcomes addressed
A student:
MGP-1
uses mathematics and statistics to compare alternative solutions to contextual problems
MGP-2
represents information in symbolic, graphical and tabular form
MGP-3
represents the relationships between changing quantities in algebraic and graphical form
MGP-9
uses appropriate technology to organise information from a limited range of practical and
everyday contexts
MGP-10
justifies a response to a given problem using appropriate mathematical terminology.
Content summary
AM1
Algebraic manipulation
AM2
Interpreting linear relationships
Terminology
algebraic expression
expand
simplify
axis
formulae
simultaneous linear equations
common difference
function
solution
constant
gradient
solve
conversion
independent variable
stepwise linear function
dependent variable
intercept
substitute
equation
linear
y-intercept
evaluate
point of intersection
Use of technology

Students will require access to appropriate technology to create graphs of functions and
to observe the effect on the graph of a function when parameters are changed.

Suitable graphing software should be used in investigating the similarities and differences
between the graphs of a variety of linear relationships.

Students could use a spreadsheet to generate a table of values and the associated graph.
Notes

Algebraic skills should be developed through practical and vocational contexts.

Application of the skills developed in this Strand in the Preliminary Mathematics General course
is consolidated further in Focus Studies in this (preliminary) course and in the HSC Mathematics
General 2 course and the HSC Mathematics General 1 course.

Students should develop an understanding of a function as input, processing, output. (It is not
intended that students learn a formal definition of a function.)
Topic Content – Algebraic Manipulation – AM1
From the Syllabus – (p52) the principal focus of this topic is to provide a
foundation in basic algebraic skills, including the solution of a variety of
equations. The topic develops techniques that have applications in work-related
and everyday contexts.
3A – Adding and Subtracting Like Terms (Back to Basics)

add, subtract, multiply and divide algebraic terms
Definition – Like terms are terms have identical pronumeral parts (the letters are exactly the same).
 THINK?
1. Does the order of the letters/pronumerals matter? Eg. Is 7ab a like term to 8ba? Why?
2. Is 3x2 a like term to 7x? Why?
3. Explain if 2y + 3y = 5y2? Why?
Exercise 3A – p80 – Q1-7 (left columns)
Extension – see below
3B – Multiplying and Dividing Algebraic Terms

add, subtract, multiply and divide algebraic terms
9y
4m
m 4wn nb
,
 5y ,


4
5n 20n
b
2w
Theory – Algebraic terms can be multiplied or divided to form single expressions. If there are algebraic fractions,
they can be simplified in the same way as number fractions.

simplify algebraic expressions involving multiplication and division, eg
Exercise 3B (p82)
Core – Q1-8 (ev 2nd part), Extension – Q9-11 (ev 2nd part)
3C – Expanding and Factorising

expand and simplify algebraic expressions
Using your brain, answer the following questions…
 THINK?
Evaluate: 5  102, 7  99, 6  57, 5  23.
Explain your mental method…
Since pronumerals represent numbers, the same law applies for algebraic expressions.
Theory
Factorising - Theory
Definition – Factorising is the reverse of expanding.
To factorise an expression, the highest common factor (HCF) needs to be taken out and placed in front of
the bracket. Divide each term inside the bracket by the HCF.
Compared to the questions above, you are given the answer and trying to find what the question was???
Example 7 - Factorising is the reverse of expanding. Factorise the following:
(a)
4m + 16
(b)
m2 + 2m
(c)
-3x – 9
Exercise 3C
Core – Q1-9,13,14 (ev 2nd part), Extension – Q10-12 (ev 2nd part)
Further Extension – see below…
E1
E2
(d)
8ab + 12bc
Diagnostic Check Up so far
1.
2.
3.
4.
5.
Simplify the following.
(a)
5m + 3m + 8
(b)
4y  5 + 3y
Simplify.
(a)
8x  4x  4 + 3x
(b)
3  7b  b + 7
(c)
t  4  3t  1
8.
Simplify:
Expand and simplify:
(a)
-5 (x + 3)
(b)
-3 (2a  4)
Expand and simplify:
(a)
 (y + 4)
(b)
 (6 + x)
(c)
2y (3y  x)
(a)
-4  2x
(b)
4m  2m
(c)
4y  (-5y2)
(d)
(-3m)  (-6m2)
(a)
4  (2  3y)
(e)
5a2  4a3
(b)
3 + 2 (x  5)
(f)
(-7x)2
Simplify:
9.
10.
Expand and simplify:
Expand and simplify:
(a)
-20x  4
(a)
3 (x + 4) + 4 (2  x)
(b)
30pq  5q
(b)
y (4  y)  3 ( 4  2y)
(c)
(-21y)  28y2
Simplify:
(a)
6.
7.
11.
3 x
x 6
(b)
5
2
m m
(c)
2 y2
y 4
(d)
4 a2
a 2
Expand and simplify:
(a)
4 (m + 3)
(b)
5 (2a  4)
(c)
4y (3  2y)
12.
Expand and simplify:
(a)
4x2 (7  x) + x (x + 1)
(b)
2a2 (a  4)  a (4a  3)
Factorising is the REVERSE of expanding.
Factorise the following expressions.
(a)
5m + 15
(b)
b2 + b
(c)
-3m – 9
(d)
6xy + 9xz
3D – Substitution

substitute numerical values into algebraic expressions, eg
3x
, 5 2x  4  , 3a2  b ,
5
5p x  y
,
yx
4m
Theory
Substitution is defined as replacement. When a number replaces a pronumeral normal calculations should apply.
Example 1
Using x = 5, y = -3 and z = 4, find the value of:
(a)
2x – z
(b)
5x + 3y
(c)
z – 3y2
3E – Substitution into Mathematical Formulas

substitute given values for the other pronumerals in a mathematical formula from a vocational or other context
to find the value of the subject of the formula,
eg if A  P 1 r  , find A given P = 600, r = 0.05, n = 3
n
Theory
–
Mathematical formulas are used to find the value of something given other variables.
D
Eg. In S = T . The single variable on the left hand side is called the ‘subject’ of the formula.
Exercise 3D p90 – Q2-5 (ev 2nd part).
Exercise 3E (p91) Q1-6 (ev 2nd part), Q7, 8, 10, 11.
Extension – see below.
Can you find similar value formulas in sports (eg NRL Supercoach)?
3F – One-step Equations

solve linear equations involving (up to) two steps, eg 5x 12  22 ,
 Theory
 To solve an equation you can:

add or subtract the same number to BOTH SIDES of an equation

multiply or divide BOTH SIDES of the equation by the same number
Examples
1.
Solve these equations.
(a)
m + 8 = 15
(b)
t  12 = 18
Use OPPOSITE
operations
+× ÷
2.
Solve these equations.
(a)
3.
5a = 35
(b)
7x = 42
Solve these equations.
(a)
4.
x2  
y
4 =3
(b)
m
6 =2
n
Check, by substituting the solution n = 117 into the equation 9 =13, to see if the answer is correct?
Exercise 3F (p93) – Q1-9 (left hand columns)
3G – Two-step Equations

solve linear equations involving two steps, eg 5x 12  22 ,
 Theory
 The process for solving two-step equations is similar to that of solving one step equations:
1. Get the furthest terms away from the pronumeral away first by doing opposite operations.
2. Then solve the remaining one-step equation.
Examples
(a)
2.
Check by substituting to see if the answer listed is the correct answer
(a)
3.
5m + 8 = 33
4y + 7 = 29
(b)
7a  6 = 57
1.
( y = 6)
(b)
43  8x = 19
(b)
15 = 9  3b
Solve:
(a)
8  4x = 3
4.
t
Solve: 4  4 = 2
5.
If v = 5m + 4, find m when v = 39
Exercise 3G (p96) – Q1, Q2(L.H.C.), Q3, Q4(a,c,e,g), Q5, Q6(LHC), Q7, Q8(LHC), Q9-11(all parts)
Extension – See below… (next handout)
( x = 4)
Two-Step Equations – Extension
E1
E2
E3
Working Space for Two Step Equation
3H – Equations with Fractions

solve linear equations involving two steps, eg
4x
x 1
r
 3,
 6 ,  3  2. \
10
3
5
 Theory
 To solve equations with algebraic fractions, multiply both sides by the Lowest Common Multiple (LCM)
of the denominators.
Examples
Solve the following for x.
1.
(a)
x 2
5=3
(b)
4 x
9 =2
2.
(a)
4x + 2
5 = -2
(b)
1
3 (4x – 1) = -3
Extension
3.
(a)
4x + 3 2
3 =5
(b)
2x – 1 2x
5 = 4
4.
(a)
2x x
5 -3 =4
(b)
x
3x
4 -2= 6
Exercise 3H (p101) – Q1-4 (every 2nd part); Extension: Q5-6 (a, c, e, and g)
End of Chapter 3 – Further Revision of topic
Pages 106-110






Review 3
3A – Review Set
3B – Review Set
3C – Review Set
3D – Review Set
Examination Questions