Linear Relationships
Master the gradient-intercept form y = mx + b, plot linear graphs, find gradients, and understand parallel and perpendicular lines.
The Gradient-Intercept Form
Every straight line on a graph can be written in the form:
y = mx + b
m = gradient (slope) — how steep the line is
b = y-intercept — where the line crosses the y-axis
Gradient (m)
The gradient tells you how much y changes for each unit increase in x.
Positive gradient: line goes up from left to right.
Negative gradient: line goes down from left to right.
Y-Intercept (b)
The y-intercept is the point where the line crosses the y-axis (where x = 0). In the equation y = 3x + 2, the line crosses the y-axis at (0, 2).
Finding the Gradient from Two Points
If you know two points on a line, (x1, y1) and (x2, y2), you can calculate the gradient using:
m = y2 − y1 x2 − x1 = rise run
Remember: "Rise over run." Rise is the vertical change (y), run is the horizontal change (x).
Plotting Linear Graphs
Below are two lines plotted on the same coordinate plane. Notice how the gradient and y-intercept determine each line's position and direction.
The blue line has gradient 2 and y-intercept 1. The red line has gradient −1 and y-intercept 2.
Parallel and Perpendicular Lines
Parallel Lines
Parallel lines have the same gradient but different y-intercepts. They never meet.
Example: y = 3x + 1 and y = 3x − 4 are parallel (both have m = 3).
Perpendicular Lines
Perpendicular lines meet at right angles (90°). Their gradients multiply to give −1.
Example: y = 2x + 3 and y = −½x + 1 are perpendicular (2 × −½ = −1).
Key Vocabulary
| Term | Definition |
|---|---|
| Gradient (m) | The steepness of a line; the rate of change of y with respect to x. |
| Y-intercept (b) | The point where the line crosses the y-axis (x = 0). |
| Linear equation | An equation whose graph is a straight line. |
| Parallel | Lines with the same gradient that never intersect. |
| Perpendicular | Lines that intersect at right angles (90°); their gradients multiply to −1. |
Worked Examples
Find the gradient and y-intercept of y = −3x + 5.
Step 1: Compare to y = mx + b.
Step 2: Identify m. The coefficient of x is −3, so the gradient m = −3.
Step 3: Identify b. The constant term is +5, so the y-intercept is (0, 5).
Interpretation: The line slopes downward (negative gradient) and crosses the y-axis at 5.
Find the gradient of the line passing through (1, 3) and (4, 9).
Step 1: Label points. (x1, y1) = (1, 3) and (x2, y2) = (4, 9)
Step 2: Apply formula. m = (9 − 3) / (4 − 1) = 6/3
Step 3: Simplify. m = 2
Interpretation: For every 1 unit increase in x, y increases by 2.
Determine if y = 4x − 1 and y = −¼x + 6 are parallel, perpendicular, or neither.
Step 1: Identify gradients. Line 1: m1 = 4. Line 2: m2 = −¼.
Step 2: Check parallel. 4 ≠ −¼, so they are not parallel.
Step 3: Check perpendicular. m1 × m2 = 4 × (−¼) = −1 ✓
Conclusion: The lines are perpendicular.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see feedback.
Question 1
What is the gradient and y-intercept of y = 2x − 7?
Question 2
What is the gradient of the line through (2, 5) and (6, 13)?
Question 3
Which line is parallel to y = 5x − 3?
Question 4
A line has gradient 3. What is the gradient of a line perpendicular to it?
Question 5
Where does the line y = −2x + 6 cross the x-axis?
Key Concepts Summary
- ● y = mx + b is the gradient-intercept form of a linear equation.
- ● The gradient (m) describes steepness; the y-intercept (b) is where the line crosses the y-axis.
- ● Gradient from two points: m = (y2 − y1) / (x2 − x1).
- ● Parallel lines share the same gradient. Perpendicular lines have gradients that multiply to −1.
- ● To find the x-intercept, set y = 0 and solve for x.