Line equations from a graph or a point
Identify gradient and intercept, then form and verify a straight-line equation.
Section Paper 2 · Extended · Non-calculator
Core and Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Read a gradient from a grid.
- Substitute coordinates and solve for a constant.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the y-intercept value of y = 4x − 3?
What you will learn
- Interpret m and c in y = mx + c.
- Find a line equation from a graph or a stated gradient and point.
- Write horizontal and vertical line equations.
Gradient controls change; intercept fixes position
In y = mx + c, m is the signed gradient and c is the y-coordinate when x = 0. The y-intercept as a point is (0, c). A line with m = 2 and c = −3 has equation y = 2x − 3: starting at −3 on the y-axis, each step of one x-unit adds two y-units.
A negative c moves the intercept below the origin; it does not make the gradient negative. For y = −3x + 5, the line falls from left to right and crosses the y-axis at five. Read the coefficient with its sign, and distinguish the intercept value from the full ordered pair.
From a plotted line
Find m from a large grid triangle and read where the line crosses the y-axis to obtain c. If the line rises three units over a horizontal change of two and crosses at y = −1, then m = 3/2 and the equation is y = (3/2)x − 1. Use coordinate values, not unscaled square counts.
Check a further clear point on the line by substitution. If (2, 2) is on this example, the equation gives (3/2) × 2 − 1 = 2. A wrong intercept or a reversed rise/run is often exposed by this extra check. A drawn estimate can only support the accuracy of the graph's scale and plotting.
From a gradient and one point
When m and a point (a, b) are supplied, substitute them into b = ma + c and solve for c. For m = −2 through (3, 1), 1 = −6 + c, so c = 7 and the equation is y = −2x + 7. The given point need not be the intercept.
Horizontal lines use y = k with gradient zero. Vertical lines use x = k and have undefined gradient. A line through (−4, 2) parallel to the y-axis is x = −4. For ordinary non-vertical lines, fully simplify coefficients and constants before giving the final equation.
Pause and explain
A line has gradient 3 and passes through (2, 5). What is c?
Worked example
A straight line has gradient −2 and passes through (3, 1). Find its equation.
Show the worked solution
- Write y = −2x + c using the stated gradient.
- Insert (3, 1): 1 = −6 + c, giving c = 7.
- The equation is y = −2x + 7; substitution of the given point verifies it.
Answer y = −2x + 7
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A plotted line has gradient 1/2 and y-intercept (0, −4). Write its equation and find y at x = 6.
Give me a hint
Insert the gradient and intercept before substituting x.
Compare my reasoning
- Use m = 1/2 and c = −4.
- The equation is y = (1/2)x − 4.
- At x = 6, y = 3 − 4 = −1.
y = (1/2)x − 4; y = −1
Look for these in your work
- Kept the intercept's sign.
- Used the equation to check a further point.
Find the line with gradient −3 passing through (−2, 5).
Give me a hint
Bracket the negative x-coordinate in the substitution.
Compare my reasoning
- Write y = −3x + c.
- 5 = −3(−2) + c = 6 + c, so c = −1.
- The line is y = −3x − 1; inserting x = −2 gives five.
y = −3x − 1
Look for these in your work
- Substituted the point correctly.
- Verified the final intercept.
A conversion graph is a straight line with gradient 2 and passes through (4, 11). Find its equation and say whether it represents direct proportion.
Give me a hint
Direct proportion must pass through the origin.
Compare my reasoning
- Use y = 2x + c and insert (4, 11).
- 11 = 8 + c, so c = 3.
- The equation y = 2x + 3 has a non-zero intercept, so it is not direct proportion.
y = 2x + 3; not direct proportion
Look for these in your work
- Derived the constant from the supplied point.
- Used the origin condition rather than just straightness.
Common mistakes
- Treating any given y-coordinate as the intercept.
- Dropping the sign of m or c.
- Forcing a vertical line into y = mx + c.
What can you explain now?
Write the equation of the vertical line through (−4, 2).
Compare with the explanation
x = −4
Every point on the vertical line keeps the same horizontal coordinate.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, find a line from its gradient and one point, then test two different points in your equation.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.