Write equations of straight lines and parallel lines
Read gradient and intercept, form a line equation from a graph or point, and use matching gradients for parallel lines.
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
In y = 2x + 3, where does the line cross the y-axis?
What you will learn
- Interpret y = mx + c and x = k
- Find a line equation from a graph or a point and gradient
- Use parallel gradients and simplify Extended line forms
Read the gradient and intercept
In y = mx + c, m is the gradient and c is the y-intercept. A vertical line is written x = k; it has no y = mx + c form.
A graph gives the y-intercept where it crosses the vertical axis. Use another point to find gradient, then check that the line equation reproduces both points.
Parallel lines share a gradient
Parallel non-vertical lines have equal gradients. Substitute the given point into y = mx + c to find the new intercept.
Core uses y = mx + c and x = k. Extended also handles ax + by = c and expects fully simplified line equations. Rearrange carefully and keep negative signs.
Pause and explain
What must be equal for two non-vertical lines to be parallel?
Worked example
Find the equation of the line with gradient −3 through (2, 1), then write 2x + y = 7 in y = mx + c form.
Show the worked solution
- Use y = −3x + c and substitute (2, 1): 1 = −6 + c, so c = 7.
- The line is y = −3x + 7.
- Rearrange 2x + y = 7 to get y = −2x + 7.
Answer y = −3x + 7; 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.
Find the equation of a line with gradient 2 and y-intercept −1, then a parallel line through (3, 5).
Give me a hint
Use y = mx + c; preserve the gradient for the parallel line.
Compare my reasoning
- The first line is y = 2x − 1.
- For the parallel line use y = 2x + c.
- Substitute (3, 5): 5 = 6 + c, so c = −1; the point lies on the first line.
Both equations are y = 2x − 1; the given point is on that line.
Look for these in your work
- Gradient and intercept read
- Parallel gradient retained
- Point checked
Rearrange 3x − 2y = 8 into y = mx + c and find a parallel line through (2, 1).
Give me a hint
Isolate y, then use its gradient and the supplied point.
Compare my reasoning
- −2y = 8 − 3x.
- y = (3/2)x − 4, so the gradient is 3/2.
- A parallel line has y = (3/2)x + c; 1 = 3 + c gives c = −2.
y = (3/2)x − 2.
Look for these in your work
- Rearrangement correct
- Gradient preserved
- Intercept found from point
A road follows y = 4x − 1 on a simplified map. A second road is parallel and passes through (1, −3). Find its equation and explain how a map reader can check it.
Give me a hint
Keep gradient 4 and solve for the intercept using the point.
Compare my reasoning
- Write y = 4x + c.
- Substitute (1, −3): −3 = 4 + c, so c = −7.
- The line is y = 4x − 7; check that it passes through (1, −3) and has the same gradient.
y = 4x − 7.
Look for these in your work
- Parallel gradient used
- Point determines intercept
- Both conditions checked
Common mistakes
- Swapping gradient and intercept
- Using y = mx + c for a vertical line
- Assuming parallel lines share an intercept
- Leaving an Extended equation unsimplified
What can you explain now?
What are the gradient and y-intercept of y = 4x − 5?
Compare with the explanation
Gradient 4; y-intercept −5
Compare coefficients with y = mx + c.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How do you identify m and c, and what stays the same when constructing a parallel line?
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.