Important Observation

Ordered pairs provide exact locations on the coordinate plane.

What Is a Linear Relationship?

A linear relationship occurs when one variable changes at a constant rate relative to another.

Example

A student earns $5 for every hour worked.

Each additional hour increases earnings by $5.

This is a linear relationship.

Equation

y = 5x

where:

• x = hours worked

• y = total earnings

Important Observation

A constant rate of change produces a straight-line graph.

Creating Tables from Equations

A table of values can be created by substituting different x-values into an equation.

Worked Example

Consider:

y = 2x + 1

Choose the x-values:

−2, −1, 0, 1, 2

Step 1: Substitute Each Value

When x = −2:

y = 2(−2) + 1 = −3

When x = −1:

y = 2(−1) + 1 = −1

When x = 0:

y = 2(0) + 1 = 1

When x = 1:

y = 2(1) + 1 = 3

When x = 2:

y = 2(2) + 1 = 5

Step 2: Organise the Values

Step 3: Write the Ordered Pairs

(−2, −3)

(−1, −1)

(0, 1)

(1, 3)

(2, 5)

These ordered pairs can now be plotted.

Creating a Graph from a Table

Worked Example

Graph the equation:

y = x + 2

Step 1: Create a Table

Step 2: Plot the Points

Step 3: Draw the Line

Use a ruler to draw a straight line through the points.

For a continuous linear relationship, extend the line appropriately.

Important Observation

The line represents all pairs of values satisfying the equation, not only the points originally listed in the table.

Recognising Linear Patterns in Tables

A linear relationship has a constant rate of change.

When x-values increase by equal amounts, the y-values change by equal amounts.

Example

The y-values increase by 3 each time x increases by 1.

Differences:

7 − 4 = 3

10 − 7 = 3

13 − 10 = 3

16 − 13 = 3

The rate of change is constant.

Equation

y = 3x + 1

Important Observation

A constant first difference indicates a linear relationship when the x-values are equally spaced.

Recognising Linear Patterns on Graphs

Linear relationships produce straight lines.

There are three common types of linear patterns.

Positive Linear Relationship

As x increases, y increases.

The line rises from left to right.

Negative Linear Relationship

As x increases, y decreases.

The line falls from left to right.

Constant Relationship

The y-value remains unchanged as x increases.

The line is horizontal.

Important Observation

The direction of a line reveals how the dependent variable changes as the independent variable increases.

Understanding Slope

The slope of a line measures its rate of change.

It describes how much y changes for a given change in x.

The slope is commonly represented by the letter m.

Formula

m = (y₂ − y₁)/(x₂ − x₁)

This can also be written as:

Slope = Rise / Run

where:

• Rise = vertical change.

• Run = horizontal change.

Worked Example

Find the slope of the line passing through:

A = (1, 3)

B = (4, 9)

Step 1: Calculate the Vertical Change

Rise = 9 − 3 = 6

Step 2: Calculate the Horizontal Change

Run = 4 − 1 = 3

Step 3: Divide

m = 6/3

m = 2

Interpretation

For every increase of 1 in x, y increases by 2.

Visualising Rise and Run

Important Observation

Slope can be positive, negative, zero, or undefined.

Positive, Negative, Zero, and Undefined Slope

Positive Slope

A line rises from left to right.

Example:

y = 2x + 1

Slope = 2

Negative Slope

A line falls from left to right.

Example:

y = −3x + 4

Slope = −3

Zero Slope

A horizontal line has slope zero.

Example:

y = 5

Slope = 0

Undefined Slope

A vertical line has an undefined slope because the horizontal change is zero.

Example:

x = 3

Important Observation

Vertical lines cannot be written in the form y = mx + b because their slope is undefined.

Slope-Intercept Form

Many linear equations can be written as:

y = mx + b

where:

• m = slope.

• b = y-intercept.

The y-intercept is the point where the line crosses the y-axis.

At the y-intercept:

x = 0

Example

Consider:

y = 3x + 2

Slope = 3

y-intercept = 2

The line crosses the y-axis at:

(0, 2)

Important Observation

The slope tells us how the line changes, while the y-intercept tells us where the line crosses the y-axis.

Graphing from Slope and y-Intercept

Worked Example

Graph:

y = 2x + 1

Step 1: Identify the y-Intercept

b = 1

Plot:

(0, 1)

Step 2: Identify the Slope

m = 2

This can be written:

2/1

Move:

• 1 unit right.

• 2 units up.

Step 3: Plot Another Point

Starting at (0, 1), moving right 1 and up 2 gives:

(1, 3)

Step 4: Draw the Line

Draw a straight line through the points.

Important Observation

A straight line is uniquely determined by two distinct points.

Additional points can be used to check accuracy.

Connecting Equations, Tables, and Graphs

A linear relationship can be represented in three main ways.

Equation

y = 2x + 3

Table

Graph

The points:

(0, 3), (1, 5), (2, 7), (3, 9), (4, 11)

lie on a straight line.

How the Representations Connect

• The equation gives the mathematical rule.

• The table shows selected input-output values.

• The graph displays the relationship visually.

Important Observation

All three representations describe the same relationship.

Finding an Equation from a Table

Worked Example

Determine the equation represented by the table.

Step 1: Find the Rate of Change

The y-values increase by 3 for every increase of 1 in x.

Therefore:

m = 3

Step 2: Find the y-Intercept

When x = 0:

y = 4

Therefore:

b = 4

Step 3: Write the Equation

y = 3x + 4

Step 4: Verify

When x = 2:

y = 3(2) + 4

y = 10

This matches the table.

Finding an Equation from Two Points

Worked Example

Find the equation of the line passing through:

(1, 4) and (3, 10)

Step 1: Calculate the Slope

m = (10 − 4)/(3 − 1)

m = 6/2

m = 3

Step 2: Use y = mx + b

y = 3x + b

Step 3: Substitute a Known Point

Using (1, 4):

4 = 3(1) + b

4 = 3 + b

b = 1

Final Equation

y = 3x + 1

Important Observation

Two distinct points determine exactly one straight line.

Interpreting Points on a Graph

Every point on a graph represents a relationship between two quantities.

Example: Distance and Time

A cyclist travels at a constant speed of 4 m/s.

The equation is:

d = 4t

where:

• d = distance in metres.

• t = time in seconds.

Table

Interpreting a Point

The point (3, 12) means:

After 3 seconds, the cyclist has travelled 12 metres.

Important Observation

A graph point should always be interpreted using the quantities and units represented by the axes.

Interpreting Lines in Real-World Situations

A line represents the overall relationship between two variables.

Example: Taxi Fare

A taxi company charges:

• A starting fee of $4.

• An additional $2 per kilometre.

The equation is:

C = 2d + 4

where:

• C = total cost in dollars.

• d = distance in kilometres.

Table

Interpretation

The slope is 2.

This means the fare increases by $2 for each additional kilometre.

The y-intercept is 4.

This represents the starting fee.

Important Observation

The slope and y-intercept often have meaningful interpretations in practical situations.

Comparing Two Linear Relationships

Graphs can be used to compare different relationships.

Worked Example

Two companies offer delivery services.

Company A:

C = 3d + 5

Company B:

C = 2d + 9

where C is cost in dollars and d is distance in kilometres.

Table

Interpretation

Company A has a lower starting fee but a higher cost per kilometre.

Company B has a higher starting fee but a lower cost per kilometre.

At 4 km, both companies charge $17.

Finding the Intersection

Set the equations equal:

3d + 5 = 2d + 9

d = 4

Substitute:

C = 3(4) + 5 = 17

Intersection = (4, 17)

Important Observation

The intersection of two lines represents a point where the two relationships have the same output value for the same input.

Linear Relationships in Science

Linear relationships are common in scientific investigations.

Example: Hooke's Law

For a spring obeying Hooke's law:

F = kx

where:

• F = force.

• k = spring constant.

• x = extension.

If k remains constant, force is directly proportional to extension.

Example

Suppose k = 20 N/m.

The graph is a straight line through the origin.

Important Observation

A straight-line graph can reveal a constant relationship between physical quantities.

Hooke's law applies only while the spring remains within its linear elastic range.

Linear Relationships in Everyday Life

Earnings

If a worker earns $12 per hour:

E = 12h

Savings

If a person begins with $50 and saves $10 each week:

S = 10w + 50

Distance

If a vehicle travels at a constant speed of 60 km/h:

d = 60t

Water Usage

If a tank contains 200 L and loses 5 L per minute:

V = 200 − 5t

Important Observation

The slope represents a rate of change, while the intercept often represents an initial value.

When Is a Relationship Not Linear?

Not every relationship produces a straight line.

Example

Consider:

y = x²

The y-values do not increase by a constant amount.

The graph is curved.

Comparison

Important Observation

A straight-line pattern indicates a linear relationship.

A curved graph generally indicates a nonlinear relationship.

Interpolation and Extrapolation

Linear graphs can be used to estimate unknown values.

Interpolation

Interpolation means estimating values between known data points.

Example

A cyclist travels:

• 8 m after 2 seconds.

• 16 m after 4 seconds.

Assuming constant speed, the distance after 3 seconds is:

12 m

Extrapolation

Extrapolation means predicting values beyond the known data range.

Example

If the cyclist continues at the same speed, the predicted distance after 6 seconds is:

24 m

Important Observation

Extrapolation assumes the linear pattern continues.

This may not always be realistic.