Grade 9 Math · Algebra
Lesson 2.1
Understanding Linear Relations
Learn what makes a relation linear, and how the rate of change and the initial value describe it.
Estimated time: 30–45 minutes
Learning goals
- I can explain what a linear relation is in my own words.
- I can find the rate of change from a table, a graph, or a description.
- I can identify the initial value of a linear relation.
- I can use the correct terms rate of change, initial value, and linear relation when I explain my thinking.
Step 1 of 7
Warm Up
A quick question to bring back what you already know.
Step 2 of 7
Learn
The idea, explained clearly — with the words mathematicians use.
What makes a relation linear
A relation connects two quantities. When the same amount is added every step, the relation is a linear relation. Plot the points and they line up perfectly straight.
Rate of change
Rate of change tells us how much one quantity changes compared with another quantity. For a bus pass that costs $4 per ride, the cost changes by $4 every time the number of rides goes up by 1. The rate of change is $4 per ride.
rate of change = change in y ÷ change in x
Initial value
The initial value is what you already have before the changes start. A gym that charges $25 to sign up has an initial value of $25, even before you attend a single month.
Putting them together
Every linear relation can be written using its rate of change and its initial value. Multiply the rate of change by the quantity that is changing, then add the initial value.
y = (rate of change) × x + (initial value)
Rate of change in everyday situations
Video slot ready · about 6 minutes. An instructional video can be embedded here when one is available.
- Linear relation
- A relationship between two quantities where the same amount is added each step, so the graph is a straight line.
- Cost of rides at $4 each: $4, $8, $12, $16.
- Rate of change
- How much one quantity changes compared with another quantity.
- Earning $15 per hour is a rate of change of $15 per hour.
- Initial value
- The value of the relation before any change happens.
- A $25 sign-up fee is the initial value of a gym membership.
- First differences
- The amounts you get when you subtract each output from the one after it. Equal first differences mean the relation is linear.
- For 5, 9, 13, 17 the first differences are 4, 4, 4.
Curriculum connection
These are the words your teacher and the Ontario curriculum use. Learn them here and use them when you explain your thinking.
- Curriculum language · C4.1Rate of change
- What this meansRate of change tells us how much one quantity changes compared with another quantity. If you earn $15 every hour, the rate of change is $15 per hour.
- Curriculum language · C4.1Initial value
- What this meansThe initial value is the amount you start with, before anything else changes. On a graph it is where the line crosses the vertical axis.
- Curriculum language · C1.4Linear relation
- What this meansA linear relation is a relationship where the same amount is added each time, so the points line up in a straight line.
Step 3 of 7
See It
Table, graph and equation — the same relationship, three ways.
The same relationship, three ways
| packages (p) | price ($) | change in price ($) |
|---|---|---|
| 0 | 40 | — |
| 1 | 46 | +6 |
| 2 | 52 | +6 |
| 3 | 58 | +6 |
| 4 | 64 | +6 |
P = 6p + 40The change of +6 in the table is the same steady climb you see in the graph, and the same 6 in front of p in the equation.
Step 4 of 7
Worked Example
Watch each step, and the reason behind it.
Worked example
A delivery job pays $40 for showing up plus $6 for each package delivered. Find the rate of change and the initial value, then write the equation.
Step 1
Find the amount added for each package. That is the rate of change.
rate of change = 6 dollars per package
Step 2
Find the pay before any packages are delivered. That is the initial value.
initial value = 40 dollars
Step 3
Multiply the rate of change by the number of packages p, then add the initial value.
P = 6p + 40
Answer: The rate of change is $6 per package, the initial value is $40, and the relation is P = 6p + 40.
Step 5 of 7
Try It
Your turn. Hints are always one tap away, and a full walkthrough is there if you need it.
Step 6 of 7
Practice
Build fluency — and see where this shows up in real life.
Real-world connection · Transit
Choosing between single fares and a monthly pass
A single transit fare costs $3.30 and a monthly pass costs $128. Riding costs form a linear relation: the rate of change is $3.30 per ride and the initial value is $0. The pass is a flat cost with a rate of change of $0 per ride.
About how many rides in a month make the monthly pass the cheaper choice?
Step 7 of 7
Mastery Check
Show what you can do now that you could not do at the start.
Quiz · unlimited attempts
Lesson Quiz
5 questions · 8 minutes
Quiz grading and saved attempts arrive in a later phase.