Do Now

Look at these three scenarios and discuss with your partner:

1. A taxi charges $5 to get in, then $2 for each kilometre
2. You walk at 4 km/h for 3 hours
   
3. A phone plan costs $40 per month

Question: Which of these would create a straight line if you graphed them? Why?

Linear Relationships in the Real World

Listen and take notes as we explore:

• What makes a relationship linear
• Graphing real-world linear equations
• Finding patterns in data

What is a Linear Relationship?

Copy this definition into your notes:

A linear relationship shows a constant rate of change

When one variable increases by a fixed amount, the other variable changes by a consistent amount too.

Key features:

• Straight line when graphed
• Constant slope (rate of change)
• Can be written as \(y = mx + c\)

Copy these notes down

Linear Equation Form: y = mx + c

• m (slope) = how much y changes when x increases by 1
• b (y-intercept) = value of y when x = 0
• Graph = straight line
• Real examples = constant rates like speed, price per item, weekly savings

Example 1: Items and Cost

Copy the scenario then fill in the table of values:

Imagine buying apples at $2 each

Let: n = number of apples C = total cost

Now use the table of values to write down the linear relationship

Example 1: Items and Cost

Study this graph and identify the pattern:

Imagine buying apples at $2 each

Equation: Cost = 2 × (Number of items)

Example 2: Distance and Time

Copy the scenario and fill in the table of values:

A car travels at 60 km/h

Let: n = number of weeks S = savings

Now use the table of values to write the linear relationship:

Example 2: Distance and Time

Compare this graph to the previous one:

A car travels at 60 km/h

Distance \(= 60 ×\)Time (in hours)

Example 3: Savings Account

Copy scenario and fill in the table of values as shown

You start with $50 and save $10 each week

Let: n = number of weeks S = savings

Now use the table of values to write the linear equation

Example 3: Savings Account

Copy scenario and draw plots as shown.

You start with $50 and save $10 each week

Equation: Savings = 50 + 10 × Weeks

Understanding y = mx + c

Copy this pattern into your notes:

Each example follows the pattern y = mx + c

Where:

• m = slope (rate of change)
• c = y-intercept (starting value)

Examples we’ve seen:

Copy these notes down

Linear Equation Form: y = mx + c

• m (slope) = how much y changes when x increases by 1
• b (y-intercept) = value of y when x = 0
• Graph = straight line
• Real examples = constant rates like speed, price per item, weekly savings

Practice problems: Give the variables names, fill in a table of values, write equation, draw graph and identify slope and y-intercept.

Think and listen

Gym membership: $30 to join + $15 per month

Let: n = number of months C = savings

Now use the table of values to write the linear equation

plot the equation. Remember to label axes

Gym membership: $30 to join + $15 per month

Equation: y = 15x + 30

Slope = 15, y-intercept = 30

Quick Check - What’s Wrong Here?

Discuss with your table - is this a linear relationship?

Exit Ticket

Complete both questions individually before you leave:

Summary

Today we learned:

• Linear relationships have constant rates of change
• They graph as straight lines
• Real examples are everywhere: costs, speeds, growth
• The equation y = mx + c describes all linear relationships
• m is the slope, c is the y-intercept

For Monday: Complete practice problems 1-6 in your workbook