Use integer-coefficient line equations and check exceptional cases
Convert between y=mx+b and ax+by=d · Read slopes and intercepts only when the relevant coefficients are non-zero · Distinguish a line from a degenerate all-plane or empty equation
Section Paper 4 · Historical Extended · calculator
Exam scope and assessment notes
Extended content; not a Core requirement. 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-intercept form
- Rearrangement and fractions
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
- Convert between y=mx+b and ax+by=d
- Read slopes and intercepts only when the relevant coefficients are non-zero
- Distinguish a line from a degenerate all-plane or empty equation
An equivalent equation must keep the same points
For ax+by=d with b≠0, rearrange to y=−(a/b)x+d/b. The slope is −a/b and the y-intercept value is d/b. Multiply every term by a non-zero common factor to clear fractions; multiplying only one term changes the line. The coefficients may all be integers even when the slope is fractional.
Check coefficients before dividing
If b=0 and a≠0, the line is vertical x=d/a. If a=0 and b≠0, it is horizontal y=d/b. At least one of a,b must be non-zero for a single line. If both vanish, 0=d means either every point when d=0 or no point when d≠0.
For 2x+3y=6, the line is y=−(2/3)x+2, with intercepts (3,0) and (0,2). Doubling all coefficients gives 4x+6y=12, the same line rather than a distinct parallel one.
Sources for this lesson
Pause and explain
What does 3x+0y=6 represent?
Worked example
Rearrange 2x+3y=6 and find its axis intercepts.
Show the worked solution
- 3y=−2x+6, so y=−(2/3)x+2.
- Set y=0 for x=3; set x=0 for y=2.
Answer y=−(2/3)x+2; intercepts (3,0) and (0,2).
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.
Write y=(1/2)x−3 with integer coefficients in ax+by=d form.
Give me a hint
Multiply the whole equation by 2.
Compare my reasoning
- 2y=x−6.
- Rearrange to x−2y=6; substitution at (0,−3) confirms equivalence.
x−2y=6.
Look for these in your work
- Clear all fractions consistently
- Check the same point set
Rearrange 3x−2y=8. State slope and axis intercepts.
Give me a hint
Isolate y without losing the negative sign.
Compare my reasoning
- y=(3/2)x−4, so slope=3/2 and y-intercept=(0,−4).
- At y=0, x=8/3.
y=(3/2)x−4; slope 3/2; intercepts (0,−4) and (8/3,0).
Look for these in your work
- Keep the sign when dividing
- Use the correct axis conditions
Classify 3x=6, 4y=8, 0x+0y=0 and 0x+0y=7. Which represent single straight lines?
Give me a hint
Check whether either variable coefficient is non-zero.
Compare my reasoning
- 3x=6 is vertical x=2; 4y=8 is horizontal y=2.
- 0=0 holds for every point, while 0=7 has none; neither is a single line.
The first two are lines x=2 and y=2; the last two are all points and no points.
Look for these in your work
- Check degenerate coefficients
- Avoid inventing a slope
Common mistakes
- Do not divide by a zero coefficient.
- Scale every term when preserving an equation.
What can you explain now?
What does 3x+0y=6 represent?
Compare with the explanation
The vertical line x=2.
Dividing the non-zero x coefficient gives a fixed x while y is free.
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Make it stick
What does 3x+0y=6 represent?
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.