What makes a relation linear?
A linear relation changes by the same amount every time. That constant amount is the rate of change. Its graph is always a straight line.
Test with first differences
For equally spaced x-values, subtract consecutive y-values. If the first differences are constant, the relation is linear.
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y | 5 | 8 | 11 | 14 | 17 |
| First differences | 3 | 3 | 3 | 3 |
Constant → linear, with rate of change 3.
Non-linear example:
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y | 0 | 1 | 4 | 9 |
| First differences | 1 | 3 | 5 |
Not constant → not linear (it’s a curve).
The equation
y = mx + b
- m = rate of change (slope)
- b = initial value (where the line crosses the y-axis)
From the first table: m = 3, b = 5 → y = 3x + 5.
Direct and partial variation
| Type | Equation | Graph |
|---|---|---|
| Direct variation | y = mx | through the origin |
| Partial variation | y = mx + b (b ≠ 0) | crosses the y-axis at b |
- Paid $16 per hour: P = 16h (direct)
- Taxi: $3.75 + $2 per km: C = 2d + 3.75 (partial)
Increasing, decreasing or constant
| m | Line | Example |
|---|---|---|
| positive | rises left to right | savings growing |
| negative | falls | phone battery draining |
| zero | horizontal | flat monthly fee |
Interpreting in context
Example: A water tank: V = 500 − 25t (litres after t minutes).
- Initial value: 500 L
- Rate of change: −25 L per minute (draining)
- Empty when V = 0: t = 20 minutes
Worked examples
Example 1: Is x: 1, 2, 3, 4; y: 10, 7, 4, 1 linear? First differences −3 each → yes; equation y = −3x + 13.
Example 2: A plumber charges $80 plus $60/h. Write the relation and classify it. C = 60h + 80, partial variation.
Example 3: Is y = 4x² linear? No: x is squared.
Common mistakes
- Using first differences when x-values aren’t equally spaced.
- Confusing the initial value with the first value in a table that doesn’t start at x = 0.
- Assuming every straight line is direct variation.
Try it yourself
- Is x: 0, 1, 2, 3; y: 2, 6, 10, 14 linear? Equation?
- Is x: 0, 1, 2, 3; y: 1, 2, 4, 8 linear?
- A car rental: $45 a day plus $0.20/km. Direct or partial?
- In y = −5x + 30, give the initial value and rate of change.
Answers: 1) yes; y = 4x + 2 2) no 3) partial 4) 30 and −5
Quick recap
- A linear relation has a constant rate of change and a straight-line graph.
- Check with first differences.
- y = mx + b: m is the rate, b is the initial value.