1. The Equation of a Straight Line
A straight line on a coordinate plane can be described by an equation that relates the x-coordinate and y-coordinate of every point on the line. The most common form is y = mx + c, where:
m is the gradient (or slope) of the line, which measures how steep the line is.
c is the y-intercept, the point where the line crosses the y-axis (where x = 0).
- Gradient (m) = change in y ÷ change in x (rise over run).
- Y-intercept (c) = the value of y when x = 0.
2. Finding the Gradient and Y-Intercept from a Graph
To find the equation of a line from its graph, follow these steps:
1. Identify two clear points on the line with exact coordinates.
2. Calculate the gradient m using the formula: m = (y₂ - y₁) / (x₂ - x₁). This gives the steepness of the line.
- Example: Points (2, 3) and (5, 9)
- m = (9 - 3) / (5 - 2) = 6 / 3 = 2
- 3. Find the y-intercept c by looking at where the line crosses the y-axis or by substituting one point into y = mx + c and solving for c.
- Example: Using point (2, 3) and m = 2, substitute into y = mx + c:
- 3 = 2 × 2 + c → 3 = 4 + c → c = 3 - 4 = -1
- 4. Write the equation: y = 2x - 1.
3. Extracting Gradient and Y-Intercept from an Equation
When given an equation in the form y = mx + c, the gradient and y-intercept can be read directly:
The coefficient of x is the gradient m. It shows how much y changes for each unit increase in x.
The constant term is the y-intercept c, the value of y when x = 0.
- Example: For y = 6x + 3, gradient m = 6, y-intercept c = 3.
- If the equation is not in this form, rearrange it to isolate y on one side.
- Example: 3y - 6x = 9 → 3y = 6x + 9 → y = 2x + 3, so m = 2, c = 3.
4. Equations of Vertical Lines
Vertical lines have an undefined gradient because the change in x is zero, which would cause division by zero in the gradient formula.
Their equations are written as x = k, where k is the constant x-coordinate of every point on the line.
These lines do not have a y-intercept because they never cross the y-axis (unless k = 0, which is the y-axis itself).
- Example: The vertical line passing through x = 4 has equation x = 4.
- This line is parallel to the y-axis and cannot be expressed in the form y = mx + c.
5. Writing Equations in Simplest Form
When providing the equation of a line, always express it in its simplest form:
For lines with a defined gradient, write as y = mx + c, simplifying fractions and combining like terms.
For vertical lines, write as x = k, where k is a simplified constant value.
- Example: If the equation is 2y = 4x + 6, divide both sides by 2 to get y = 2x + 3.
- Example: If the gradient is a fraction like 4/2, simplify it to 2 before writing the equation.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Identify two points | Choose points on the line with exact coordinates | (1, 2) and (3, 6) | Ensures accurate gradient calculation |
| Calculate gradient | Use m = (y₂ - y₁) / (x₂ - x₁) | m = (6 - 2) / (3 - 1) = 4 / 2 = 2 | Marks awarded for correct formula and substitution |
| Find y-intercept | Substitute one point into y = mx + c and solve for c | 2 = 2 × 1 + c → c = 0 | Show clear working to gain method marks |
| Write equation | Use y = mx + c with found m and c | y = 2x + 0 → y = 2x | Simplify and write final answer clearly |
| Recognise vertical lines | Equation is x = k, no y-intercept | x = 5 | State equation in simplest form |
Active recall
Quick check
1. What is the gradient and y-intercept of y = 4x - 7?
Gradient m = 4, y-intercept c = -7.
2. Find the equation of the line passing through points (1, 3) and (3, 11).
Gradient m = (11 - 3) / (3 - 1) = 8 / 2 = 4; using (1, 3): 3 = 4 × 1 + c → c = -1; equation y = 4x - 1.
3. Write the equation of the vertical line passing through x = -2.
Equation is x = -2.
4. Given y = (2/3)x + 5, what is the gradient and y-intercept?
Gradient m = 2/3, y-intercept c = 5.