Module Three: Linear and Quadratic Skills - Mathematics Pathways

Module Three: Linear and Quadratic Skills

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.

Module Three:  Pre-test

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:

10 Ways Simultaneous Equations Can Be Used in Everyday Life

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:

101 uses of a quadratic equation: Part II

CIMT - The Co-ordinate System

Learn More

Maths is Fun - Cartesian Coordinates

Khan Academy - The Co-ordinate Plane

Lesson 2 - Introduction to linear equations

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

Lesson 3 - Introduction to linear graphs

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.

CIMT - Plotting Graphs from their Equations

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

CIMT - Plotting Graphs from their Equations

To cover any areas where you need more experience.

 

Lesson 4 - More Linear Equations

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

Lesson 5 - Graphing Linear Equations

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

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.

CIMT - Plotting Graphs Given Their Equations

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.

CIMT - Plotting Graphs Given Their Equations

Lesson 1 - Simultaneous Equations

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

This interactive worksheet and self test will help to check your skills and fill in any gaps.

CIMT - Simultaneous Equations

Learn More

This animated tutorial at Mathtutor explains the concept of simultaneous equations.

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

Lesson 1 - What is a quadratic function

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.

 
Image source : http://www.cimt.plymouth.ac.uk/projects/mepres/book9/bk9i17/bk9_17i2.html

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.

Math Is Fun - Maths Resources - Quadratic Equations

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.

 

 

Math Is Fun - Maths Resources - Quadratic Equations

Lesson 2 - Solving quadratic equations by factorising

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/

Lesson 3 - Solving quadratic equations by using the formula

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 
 

The part that says + / - means there are actually two versions to be calculated. First with a plus sign and then with a minus sign. That is how you get the two roots

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

Watch the Khan Academy Lesson on The Quadratic Formula

 

Use the Math Is Fun - Maths Resources for Quadratic Equations

 All Khan Academy content is available for free at www.khanacademy.com

 

Comprehensive set of tutorials and tests

Detailed video lessons of around 20 to 40 minutes, PDF text books, diagnostic tests and exercises 

A series of video tutorials on a range of mathematics topics.

Has a good set of interactive explorers under the Tools heading.

When you have completed the module, check your knowledge with this multiple choice post-test. 

• This test will confirm your level of understanding of the skills in this module.
• Please attempt all questions on the test.
• To achieve 'mastery' (the level at which you are truly competent with these skills) you will need to achieve at least 80%
• You will need to enter your name and email address in the test to have the results emailed to you.

• If you want to use these results to apply to study business at UTAS or UNDA, ensure that you save them to use as evidence that you have passed the module.

Module Three: Post-test