Every straight line has a constant steepness called its gradient. Once you know the gradient and one point on the line, you can write its equation in three different forms — and each form is useful in different situations.
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What you need to know
How to find the gradient between two points using the formula
The three forms of a straight line equation: y = mx + c, the point-gradient form, and the general form
How to find the equation of a line given a gradient and a point, or two points
How to read the x-intercept and y-intercept from each form
How to rearrange between forms — especially clearing fractions for the form ax + by + d = 0
Finding the Gradient
The gradient tells you how steep the line is. To find it, pick any two points on the line — call them (x1, y1) and (x2, y2).
m = y2 − y1x2 − x1
(given in the formula booklet)
Gradient = rise ÷ run
The line goes up by y₂ − y₁ across a horizontal distance of x₂ − x₁
What the gradient tells you
The number you get from the formula tells you how the line behaves:
Positive
m > 0 — going up
Negative
m < 0 — going down
Zero
m = 0 — horizontal
Undefined
vertical line
Quick read: a gradient of 1 means up 1, right 1. A gradient of −2 means down 2, right 1. The bigger the absolute value, the steeper the line.
The Three Forms of a Straight Line
A straight line can be written in three equivalent forms. Each one is best for a different situation.
Gradient-Intercept
y = mx + c
Best when you can read the gradientm and the y-intercept(0, c) directly.
Point-Gradient
y − y1 = m(x − x1)
Best when you have a gradient and a known point(x1, y1).
General Form
ax + by + d = 0
Used when answers must be in integer form. Both intercepts are easy to find.
From the general form, the x-intercept is −da and the y-intercept is −db. Set the other variable to 0 and solve.
Finding the Equation of a Line
Whatever information you’re given, follow this strategy:
If you have two points, find the gradient first using the formula.
Plug the gradient and any one point into the point-gradient form: y − y1 = m(x − x1).
Rearrange into the form the question asks for — usually y = mx + c or ax + by + d = 0.
For general form, multiply through to clear fractions and ensure a, b, d are integers.
GDC shortcut: on Paper 2, enter both points in statistics mode and run a linear regression. The calculator gives you y = ax + b directly.
Worked Examples
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Example 1 — Find the gradient between two points
Find the gradient of the line passing through the points (1, −3) and (5, 9).
Answer:
Step 1: label the points and apply the formula.(x₁, y₁) = (1, −3), (x₂, y₂) = (5, 9)m = (9 − (−3)) / (5 − 1)m = 12 / 4m = 3Positive gradient — the line goes up from left to right.
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Example 2 — Find the equation given gradient and a point
A line has gradient 4 and passes through the point (2, −3). Find its equation in the form y = mx + c.
Find the equation of the line passing through (−1, 4) and (3, 12). Give your answer in the form y = mx + c.
Answer:
Step 1: find the gradient.m = (12 − 4) / (3 − (−1))m = 8 / 4 = 2Step 2: use point-gradient form with (−1, 4).y − 4 = 2(x − (−1))y − 4 = 2(x + 1)y − 4 = 2x + 2y = 2x + 6Check: at x = 3, y = 2(3) + 6 = 12 ✓
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Example 4 — Equation in general form (integer coefficients)
The line l passes through the points (−2, 5) and (6, −7). Find the equation of l in the form ax + by + d = 0, where a, b, and d are integers.
Answer:
Step 1: find the gradient.m = (−7 − 5) / (6 − (−2))m = −12 / 8 = −3/2Step 2: use point-gradient form with (−2, 5).y − 5 = −3/2 (x − (−2))y − 5 = −3/2 (x + 2)Step 3: multiply both sides by 2 to clear the fraction.2(y − 5) = −3(x + 2)2y − 10 = −3x − 6Step 4: rearrange so right side is 0.3x + 2y − 10 + 6 = 03x + 2y − 4 = 0a = 3, b = 2, d = −4 — all integers ✓
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Example 5 — Find the x-intercept and y-intercept
The line l has equation 3x + 4y − 12 = 0. Find the coordinates of the x-intercept and the y-intercept.
Answer:
Step 1: x-intercept — set y = 0.3x + 4(0) − 12 = 03x = 12 → x = 4x-intercept: (4, 0)Step 2: y-intercept — set x = 0.3(0) + 4y − 12 = 04y = 12 → y = 3y-intercept: (0, 3)
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Tips
Always start with point-gradient form. It works for every situation — just plug in and rearrange.
Pick the easier point. If you have two points, pick the one with smaller or simpler numbers to avoid sign errors.
Check your answer: plug a known point back into your final equation. Both sides should match.
For general form, if your gradient has a denominator, multiply both sides by it to clear the fraction.
On Paper 2, use your GDC’s linear regression mode for instant equations from two points.
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Common mistakes
Mixing up the order in the gradient formula. The y-difference goes on TOP, the x-difference on the bottom. y₂ − y₁x₂ − x₁ — never the other way around.
Sign errors with negative coordinates. Subtracting a negative becomes adding: y − (−3) = y + 3, not y − 3.
Wrong order of subtraction. Keep the same order on top and bottom — if you do (y2 − y1) on top, it must be (x2 − x1) on the bottom — same direction.
Forgetting to clear fractions when the question asks for integer form ax + by + d = 0. If your gradient is a fraction, you must multiply through.
Stopping too early. If the question wants y = mx + c, you must finish the rearrangement — don’t leave it as y − 5 = 2(x + 1).
Confusing the y-intercept with the gradient. In y = 3x + 7, the gradient is 3 and the y-intercept is (0, 7) — not the other way round.
Final word: Master the gradient formula and the point-gradient form, and you can answer any straight-line question. Everything else is just rearranging.
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