Find a line equation and test it against all given points
Find slope from two distinct points · Use one point to calculate the intercept · Reject a third point that does not satisfy the proposed line
Section C7.5 · Core · Straight-line equations
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Slope and y=mx+b
- Substitution
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Find slope from two distinct points
- Use one point to calculate the intercept
- Reject a third point that does not satisfy the proposed line
Slope first, intercept second
For two points with different x coordinates, first calculate m, then use b=y−mx from either point. For (1,3) and (3,7), m=4/2=2 and b=3−2=1, giving y=2x+1. Check both original points by substitution. Two different intercept calculations must agree if the arithmetic is correct.
More points are more constraints
A line through two distinct points need not pass through every extra listed point. Substitute each additional pair. For (0,1),(2,5), the line is y=2x+1; at x=3 it requires y=7, so (3,8) is not on it. Repeated identical points are not enough to determine a unique line, and a vertical pair cannot supply a finite slope-intercept equation.
Sources for this lesson
Pause and explain
After finding m, how can b be found from a point (x,y) on the line?
Worked example
Find the line through (1,3) and (3,7).
Show the worked solution
- m=(7−3)/(3−1)=2.
- b=3−2×1=1; check 7=2×3+1.
Answer y=2x+1.
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 line through (−1,5) and (2,−1).
Give me a hint
Calculate slope and then the intercept.
Compare my reasoning
- m=(−1−5)/(2+1)=−2.
- b=5−(−2)(−1)=3; both points satisfy the line.
y=−2x+3.
Look for these in your work
- Use signed differences
- Verify both endpoints
Find the line through (0,−2) and (4,0).
Give me a hint
One point already gives the y-intercept.
Compare my reasoning
- b=−2; m=(0+2)/(4−0)=1/2.
- The equation y=(1/2)x−2 gives both points.
y=(1/2)x−2.
Look for these in your work
- Read the intercept correctly
- Check the slope
Can one straight line contain (0,1), (2,5) and (3,8)? Explain using the first two points.
Give me a hint
Find their line and test the third.
Compare my reasoning
- The first two determine y=2x+1.
- At x=3 this gives 7, not 8.
No; the third point fails y=2x+1.
Look for these in your work
- Test all supplied data
- Reject an inconsistent model
Common mistakes
- Check the denominator before calculating slope.
- Two fitting points do not establish that a third fits.
What can you explain now?
After finding m, how can b be found from a point (x,y) on the line?
Compare with the explanation
b=y−mx.
Rearrange the line equation and substitute that point.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
After finding m, how can b be found from a point (x,y) on the 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.