91029 Linear Algebra

In this unit you will make connections between different representations for linear problems. Using these representations you will learn to solve problems.

These skills are a core foundation for all mathematics.

Introduction

Algebra is all around us at all times, but it is not limited to just numbers and equations. You will likely have thought through algebraic problems without recognising it as such.

In this unit we will learn how to work with various ways that algebra might present itself.

Words

In the real world, a lot of problems are presented in context using words.

To help you learn about the different representations, this webpage will use the same problem for you to compare.

Problem

You are hosting a lunch with friends. You decide to get sushi with customised ingredients, and are comparing the costs of three different companies:

Authentic Sushi (A)

Each sushi will cost $0.50. You will also need to pay an extra $15 for making a custom order.

Buffet Sushi (B)

Pay $70 to get as many custom sushi as you like!

Custom Sushi (C)

Pay $0.75 for every custom sushi. There is no further costs.

Which company should you order from? This will depend on how many pieces of sushi you need. Let's represent this problem in different ways to help answer this question.

Tables

An easy way to help understand the problem is by setting up a table.

Questions for you:

Can you complete the table?
Can you extend the table further?
Hypothetically, what would you be paying if you ordered zero sushi?
Which sushi shop should you go with if you wanted to order 20 pieces of sushi?

Important Connections to other representations

Y-intercept

We find this by finding the hypothetical cost when we order zero sushi.

Gradient

We find this by looking at the "per sushi" cost (ie how much does the cost change when we add 1 extra sushi?)

Graphs (with Tables)

While tables are useful, they can be tedious to set up and slow to read. A graph is a more visual representation that can provide a lot of information at a glance!

We can draw a table by hand. If you do, use rulers and space the values correctly.
Alternatively, we can use an online graphing tool such as Desmos.

Using Desmos

Add a Table to Desmos

2. Resize the screen

3. Find your values

This is nice, but we still need to manually enter each value into a table. We need a way to describe the problem in a brief mathematical equation!

Important Connections to other representations

Plotted values for Authentic Sushi

Y-intercept

We find this by finding the cost for zero sushi. In this case, it is $15.

Gradient

Each time we go across 1 sushi, we go up in price by $0.50. So our gradient is $0.50.

Forming an Equation

By setting up an equation, we can quickly find the cost for any number of sushi.

There are a few different ways to set up the equation.

Here is one of them:

We short-hand this as:

where m is the gradient, and c is the y intercept. x represents the number of sushi, and y represents the cost.

We always leave x and y as letters, but need to find the numeric values of the gradient (m) and the y-intercept (c).

How do we find these values? See the connections we made in the table and graph sections!

For Authentic Sushi, we learn that the gradient is $0.50 and the y-intercept is $15!

So we just replace m with 0.50 and c with 15:

Solving Equations

An equation tells us how the cost changes as we add more sushi to the order.

Solving equations requires you to use a mix of algebraic skills. Refer to this page of the maths website to learn and work through various algebraic skills.

Graphs (with Equations)

We can enter equations into Desmos to create the entire graph, rather than point by point.

This graph now shows the cost for any number of sushi!

If you enter the equation for each sushi company, we can compare the 3 graphs to find which is cheaper, depending on how many sushi you want to order.

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