Equations of Straight Lines (y = mx + c)
Understanding y = mx + c
Every straight line can be described by the equation y = mx + c, which is called the slope-intercept form. This simple formula contains all the information you need to understand and draw a line.
m = gradient (or slope) – tells you how steep the line is. It's the rise divided by the run: how many units the line goes up (or down) for every 1 unit it moves right.
c = y-intercept – the exact point where the line crosses the y-axis. This is the y-coordinate when x = 0.
Reading m and c from an Equation
Simply look at the equation and identify the coefficient of x (that's m) and the constant term (that's c):
✓ Examples
y = 3x + 5: m = 3 (gradient), c = 5 (y-intercept)
y = −2x + 7: m = −2 (gradient), c = 7 (y-intercept)
y = 4x − 1: m = 4 (gradient), c = −1 (y-intercept)
y = −0.5x + 3: m = −0.5 (gradient), c = 3 (y-intercept)
Finding an Equation from a Graph
When you have a graph, you need to find m (the gradient) and c (the y-intercept) separately, then write them into y = mx + c.
Step 1: Find the y-intercept (c)
Look at where the line crosses the y-axis. Read off the y-value.
Step 2: Find the gradient (m)
Pick two clear points on the line. Draw a right-angled triangle between them. Calculate rise ÷ run. Remember: positive if uphill, negative if downhill.
Step 3: Write the equation
Substitute your values of m and c into y = mx + c.
✓ Finding an Equation from a Graph
From the graph, read: y-intercept = 5, and the line passes through (2, 4) and (10, 0).
Find the gradient:
rise = 4 − 0 = 4, run = 2 − 10 = −8
m = 4 ÷ (−8) = −1/2 (negative because downhill)
Equation: y = −(1/2)x + 5
Try This
What is the gradient and y-intercept of the line y = 6x − 3?
Finding an Equation When You Know a Gradient and a Point
If you know the gradient (m) but not the y-intercept (c), you can find c by substituting a point you know the line passes through.
Step 1: Write y = mx + c with your known m value
Step 2: Substitute the point (x, y) into the equation
Step 3: Solve for c
Step 4: Write the complete equation
✓ Example: Gradient and Point
A line has gradient 3 and passes through (5, 4). Find its equation.
Step 1: Write y = 3x + c
Step 2: Substitute x = 5, y = 4
4 = 3(5) + c
4 = 15 + c
Step 3: Solve: c = 4 − 15 = −11
Step 4: y = 3x − 11
Try This
Find the equation of a line with gradient −2 that passes through (3, 1).
Special Cases: Horizontal and Vertical Lines
- Horizontal lines: y = c (no x term, gradient = 0). Example: y = 5
- Vertical lines: x = k (can't use y = mx + c format, gradient is undefined). Example: x = −2
Rearranging Equations to Standard Form