UTAS Home › › Mathematics Pathways › Pathways to Business › Module Three: Linear and Quadratic Skills
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Click on the link below to take the Pre-test for Module Three. You will be given 5 random questions to test your knowledge of linear and quadratic skills.
If you receive more 80% or more, you can move onto the next module. If your score less than 80%, work through the module lessons and take the post-test at the end of the module.
Make sure you enter your full name and email address so your results can be emailed to you. You will need to print out or save these results for your records. You may need to show them to your university.
Applications of Linear and Quadratic Equations in Business
Linear Equations are used:
To calculate the Break Even Point for a product using the relationship:
Total Cost = Fixed Cost + Variable Cost
To analyse or predict profit:
Profit = Income - Total Cost
For example Tom's Tyres who manufacture and sell tyres have obtained the following information on a new tyre product they would like to manufacture.
Total Fixed Costs $40 000
Sales price per unit $100
Variable Costs per unit $60
They want to calculate the breakeven point in units for the new product. The Break Even Point (BEP) can be calculated using the formula (linear equation):
BEP = Fixed costs/ (Sales Price per unit – Variable Costs per unit).
Thus in this example BEP = $40 000/ ($10 - $6) = 10 000 units
Linear equations can also help in situations where you have known quantities of raw materials, and need to calculate how much finished product can you make?
Simultaneous Equations
Simultaneous equations are helpful where there are two or more unknowns in a problem and you need to know two or more sets of information about the problem.
For example:
Simultaneous equations could be used to investigate the relationship between price and likely sales of a product. That is, how price sensitive is the market?
Comparison of two scenarios with different pricing such as two photocopier purchases which has an upfront cost plus a running cost, could be undertaken using simultaneous equations.
Comparing two or more scenarios for investing to grow a business. Do you spend more on marketing or in improving plant and equipment? How do you get the best balance?
The following websites provide some examples of the uses of equations:
Quadratic Equations
Quadratic equations can help with problems such as optimisation. You may want to design a container that holds a known amount of product but uses the least amount of raw materials.
Any calculations involving speed, acceleration, trajectories or falling objects will involve quadratic equations.
Examples of the uses of quadratic equations can be found at the
101 uses of a quadratic equation site:
In this topic you will revise the following
Lesson 1 - The co-ordinate system
Lesson 2 - Introduction to linear equations
Lesson 3 - Introduction to linear graphs
Lesson 4 - More Linear Equations
Lesson 5 - Graphing Linear Equations
Note: Linear Functions and Linear Equations
A linear function is a way of showing the relationship between two or more variables such as:
Example
A book costs 6 dollars and no matter how many books you buy the delivery cost is $12
y = 6x + 12
You can find a y value (total cost) for any x value (number of books).
A linear equation is the statement that allows us to calculate one variable when we know the other.
If the total bill was $30, how many books were bought.
So 3 books were bought.
In many Maths resources, the terms Linear Equation and Linear Function are used interchangeably so don’t get confused by this.
About This Lesson
The co-ordinate system, also known as the Cartesian Co-ordinate system is used to represent mathematical relationships graphically. In this module we are concerned with writing one variable as a function of another so we use a 2 dimensional x y plane like this:
Test Yourself
This interactive worksheet and self test will help to check your skills and fill in any gaps.
CIMT - The Co-ordinate System
Learn More
If you need help with this area, this is a simple introduction
Maths is Fun - Cartesian Coordinates
This Khan Academy series include tutorials and a self tests.
Khan Academy - The Co-ordinate Plane
About This Lesson
The term linear means the power of the variables is 1. There are no squares, cubes or any higher powers and no roots or fractional powers.
It is called a linear equation because its graph of its function is a straight line.
In this lesson you will revise simple equations which do not involve a great deal of transposition
Example
What is x if
x+7 = 12
Just by looking at the question you can see that x = 5 will give the correct answer.
In this case the answer is easy, in some cases you will need to do some transposition of terms to work out the answer
This interactive Linear Function Explorer from Math Open Reference will help you test your understanding.
Image source: http://www.mathopenref.com/linearexplorermxb.html
Test Yourself
This interactive worksheet and self test will help to check your skills and fill in any gaps.
CIMT - Linear Equations
Learn More
Khan Academy - The Why of Algebra
Khan Academy - Equations for Beginners Series
Khan Academy - Linear Equation Word Problems
About This Lesson
Linear graphs are represented by straight lines on the co-ordinate plane
y = 2x + 1 looks like this
Image source: http://www.cimt.plymouth.ac.uk/projects/mepres/book8/bk8i14/bk8_14i3.htm
If you know an equation, you can draw its graph and vice versa.
Test Yourself
This interactive worksheet and self test will help to check your skills and fill in any gaps.
Learn More
Some simple explanations can be found at
Math Is Fun - Maths Resources - Explore the properties of a straight line graph
More comprehensive tutorials at
Khan Academy - Graphing Solutions to Equations
Use the built in help in
To cover any areas where you need more experience.
About This Lesson
This is a continuation of the work in Lesson 2 - Linear Equations
The major difference is that the equations are more complex and require more manipulation to solve. The variable may appear more than once or on both sides of the equation.
Example
6x - 2 = 4x + 8
Test Yourself
This interactive worksheet and self test will help to check your skills and fill in any gaps.
CIMT - Linear Equations 2
Learn More
Khan Academy - Variables on Both Sides
Khan Academy - Super Yoga Plans explains linear equations using the example of comparing Yoga class fee plans.
Work through the CIMT tutorial in detail using the built in help.
CIMT - Linear Equations 2
About This Lesson
There are a number of ways of showing a linear equation as a graph.
By plotting points on the graph
Image source: http://www.cimt.plymouth.ac.uk/projects/mepres/book8/bk8i14/bk8_14i3.htm
By using the slope of the graph
And by using the x and y intercepts on the axes.
In this lesson we will revise these methods.
Test Yourself
This interactive worksheet and self test will help to check your skills and fill in any gaps.
Learn More
Khan Academy - Graphing by Plotting Points
Khan Academy - Graphing Using x and y intercepts
Khan Academy - Graphing Using Slope (Gradient)
Work through the CIMT tutorial again using the built in help.
In this topic you will revise the following
Lesson 1 - Simultaneous Equations
Simultaneous equations are used for....
About This Lesson
Image source: http://www.mathtutor.ac.uk/algebra/simultaneouslinearequations/animation2
Simultaneous equations refers to situations where we have two or more equations about the same problem which are both true at the same time.
Example
If you buy 4 coffees and 3 teas and the total cost is $18, that is not enough information to work out the cost of one coffee. There are a number of possibilities ranging from
Coffee $4.50 and Tea is free (unlikely but possible)
To
Coffee is free and Tea $6 (also unlikely but possible)
The correct answer would be somewhere in between.
If you go back the next day and buy 3 coffees and 2 teas and the total cost is $13 then you have more information and can work out the correct cost of each.
Looking at these as equations where C is the cost of one coffee and T is the cost of one tea:
Day 1
4C + 3T = 18
Day 2
3C + 2T = 13
With a bit of trial and error you can see that Coffee $3 and Tea $2 is the only solution that works in both cases. Try it and see.
This is a simple example but you can use algebraic methods to solve much more difficult problems of this type.
There are three methods of solution:
Substitution of one variable for another.
Elimination of one of the variables
Graphing the two equations to see where they intersect.
Check your answer here
All these methods are covered below.
Test Yourself
CIMT - Simultaneous Equations
Learn More
Mathtutor - Simultaneous Equations animation
If you need more practice, go back to
CIMT - Simultaneous Equations
And use the "Show me" links to see the three ways of solving simultaneous equations then try the exercises at the end.
This Mathtutor series has a 35min video lesson plus exercises and a printable PDF which covers the topic well.
Mathtutor - Simultaneous Equations Lessons
Direct link to the printable PDF on Simultaneous Linear Equations
In this topic you will revise the following
Lesson 1 - What is a quadratic function
Lesson 2 - Solving quadratic equations by factorising
Lesson 3 - Solving quadratic equations by using the formula
About This Lesson
A quadratic equation is one where the highest power of the variable is 2, for example
x2+2x - 3 = 0
A quadratic equation will generally have two roots, in other words there are two values of x which make the above statement true. In special cases, a quadratic equation can have no roots or two roots which are the same, so effectively one root.
Finding the roots of quadratic equations is an important mathematical skill and there are several ways of doing it. There is more about that in later lessons.
Here is a graph of the quadratic function x2+2x - 3 = y which shows the values of y for various values of x.
To solve the equation x2+2x - 3 = 0 we need to look at the points where y = 0 that is where the graph crosses the horizontal axis. You can see that is where
x = 1 and x = -3 and these are the roots of the equation.
The shape of the graph is called a parabola and in this case there is an axis of symmetry at
x = -1
Hidden Quadratics
Sometimes an equation may not look like a quadratic equation for example:
3x(2x-1) = 6
but when simplified and with all terms moved to the left it becomes
6x2 - 3x - 6 = 0
Test Yourself
The link below covers most of the work in this module. If you can understand all of it and the questions at the end, you can skip the lessons in this module.
Learn More
Spend some time with this Quadratic Equation Explorer from Math Open Reference. It allows you to change the values of a, b and c in the equation ax2 + bx + c = 0 to create different equations and see the resulting graph of ax2 + bx + c = y
Image source: http://www.mathopenref.com/quadraticexplorer.html
This video tutorial from Maths Gives You Power is an overview of quadratic equations.
About This Lesson
Note: that in the US they use the term factoring rather than factorising
Here is a graph of the quadratic function x2+2x - 3 = y from Lesson 1
Image source : http://www.cimt.plymouth.ac.uk/projects/mepres/book9/bk9i17/bk9_17i2.html
To solve the equation x2+2x - 3 = 0 by factorising you need to turn the expression on the left into a pair of factors multiplied together.
That is (x + 3)(x - 1) =0
This allows us to find the roots because of the Null Factor Law which tells us that
If ab = 0
Then a must equal zero or b must equal zero or both must equal zero.
So either x + 3 = 0
therefore x = -3
Or x - 1 = 0
Therefore x = 1
So these are the roots of the equation which agrees with what you see from the graph.
Test Yourself
CIMT - Quadratic Equations by Factorising
Learn More
Watch the video below from Maths Gives you Power - Quadratic Equations by Factoring Part 1
Khan Academy Lessons on - Solving Quadratics by Factoring
Look at Math Is Fun - Maths Resources - Quadratic Equations
All Khan Academy content is available for free at http://www.khanacademy.org/
About This Lesson
Sometimes it is not possible to solve a quadratic equation by factorising no matter how hard you try and graphing does not always give an exactly precise value so another method is needed.
This is the Quadratic Formula Method which will work for any quadratic equation that has roots.
This is the formula. It looks complicated and needs to be worked through carefully.
When ax2+ bx +c = 0 then the roots are
Test Yourself
Math Is Fun - Maths Resources - Quadratic Equations Answer the questions at the end of the page
Learn More
Watch the following video from Maths Gives you Power - Solving quadratic equations by using the formula
All Khan Academy content is available for free at www.khanacademy.comWatch the Khan Academy Lesson on The Quadratic Formula
Use the Math Is Fun - Maths Resources for Quadratic Equations
Khan Academy - Algebra 1
Comprehensive set of tutorials and tests
Detailed video lessons of around 20 to 40 minutes, PDF text books, diagnostic tests and exercises
Maths Gives You Power - YouTube Channel
A series of video tutorials on a range of mathematics topics.
Math Open Ref - Index Page
Has a good set of interactive explorers under the Tools heading.
All Khan Academy content is available for free at www.khanacademy.com
When you have completed the module, check your knowledge with this multiple choice post-test.
PLEASE NOTE:
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2 May, 2018
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