Line equations from a graph or a point
Identify gradient and intercept, then form and verify a straight-line equation.
Core and Extended.
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
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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons; Geometry contains 15. Coordinate geometry and Geometry each have a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other five syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice and paper constructions are self-checked, not automatically graded. The checkpoints sample skills and do not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources