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2. Linear Relations and Equations
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AustraliaVIC
VCE 11 General 2023

2.01 Linear relations

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Lesson

Introduction

A relationship between two variables is linear if:

  • a linear equation can be used to relate the two variables

  • the graph of the relationship on a number plane is a straight line

  • the dependent variable changes by a constant amount as the independent variable changes

Recognise linear relationships from a graph

When shown a graph that represents a relationship between two variables, if the graph is a straight line, then the graph is a linear relation and a linear equation can be used to represent the relationship. Below is an example of a linear relation:

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y

In this chapter, linear graphs will be created using technology, such as a CAS calculator. Learning how to sketch linear graphs will be covered in more detail in Chapter 9.

Examples

Example 1

Consider the graph plotted below.

-10
-5
5
10
x
-10
-5
5
10
y

Is the equation of this graph linear?

A
Yes
B
No
Worked Solution
Create a strategy

Check if y values on the graph change by a constant amount for every unit increase in x.

Apply the idea

Consider the meaning of "linear". Based on the graph, it is not a straight line. Moreover, the values of y do not change by the constant amount for every unit increase in x.

So, the equation of the graph is not linear. The correct answer is Option B.

Idea summary

When shown a graph that represents a relationship between two variables, if the graph is a straight line, then the graph is a linear relation and a linear equation can be used to represent the relationship.

Recognise linear relationships from an equation

An equation is linear when the equation is or can be arranged into one of the following forms:

  • y=ax+b, where y (the dependent variable) is the subject of the equation

  • Ax+By = C where A,B,C are constants

The x term in a linear relationship will always have a power of 1, though the power is rarely written explicitly.

Examples

Example 2

y is equal to 7 less than 2 lots of x.

a

Write the statement above as a mathematical equation.

Worked Solution
Create a strategy

Remember that 'less than' means to subtract.

Apply the idea

We want to form an expression for "2 lots of x" and subtract 7. This will then be equal to y.

So, the mathematical equation of the statement above is given by:y=2x-7

b

Is this equation linear?

A
Yes
B
No
Worked Solution
Create a strategy

Consider that if x and y have linear relationship, then their equation is of the form y=a+bx.

Apply the idea

Based on the result found from part (a), the equation is of the form y=a+bx, where a=-7 and b=2. This means that the equation is linear.

So, the correct answer is Option A.

Idea summary

Linear equation forms:

  • y=ax+b, where y (the dependent variable) is the subject of the equation

  • Ax+By = C where A,B,C are constants

Recognise linear relationships from a table of values

When determining a relationship between two variables, a table of values can be used to display several values for a given independent variable (x) with corresponding values of the dependent variable (y).

A table of values such as the one above can be used to recognise a linear relationship, if there is a common difference between y values as x changes by a constant amount.

If the table has consecutive x values: for each 1 unit change in x, if the y value changes by the same amount each time then the relationship between x and y must be a linear relation (i.e. the gradient is always the same).

Consider the table of values given below. Does the table represent a linear relation?

x3456
y12151821

Some things to note about this table: the x values go up by 1 each time and it doesn't matter what number the table starts at.

Notice in the above table that, for each 1 unit increase in x, y increases by 3. This is a linear relationship, as this constant change in 3 indicates a common difference.

So we can check for a linear relationship by looking for a common difference between the y values. If each successive y value has the same difference then it is linear.

Examples

Example 3

Would the following table of values represent a linear graph?

a
x3691215
y-7-14-21-28-35
A
Yes
B
No
Worked Solution
Create a strategy

Check if for constant increases in x, y changes by a constant amount.

Apply the idea

From the above table, notice that the x values go up by 3 each time. For each 3 unit increase in x, y decreases by 7.

So, the given table represents a linear graph as there are constant changes in the values of x and y values. The answer is Option A.

b
x721354963
y\dfrac{15}{2}15\dfrac{45}{2}45\dfrac{135}{2}
A
No
B
Yes
Worked Solution
Create a strategy

Check if for constant increases in x, y changes by a constant amount.

Apply the idea

From the above table, notice that the x values go up by 14 each time. For each 14 unit increase in x, the y values do not change by a constant amount.

So, the given table does not represent a linear graph and the answer is Option A.

Idea summary

We can check for a linear relationship between the two variables by looking for a common difference between the x and y values. If for each successive x and y values have the same difference then it is linear.

Outcomes

U1.AoS4.1

the properties of linear functions and their graphs

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