Solve and interpret linear systems graphically
Read a shared solution from two line graphs · Distinguish intersecting, parallel and coincident lines · Check graphical estimates algebraically
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
- Line equations and slope
- Algebraic linear systems
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
- Read a shared solution from two line graphs
- Distinguish intersecting, parallel and coincident lines
- Check graphical estimates algebraically
Intersection means both conditions
Convert each equation into a suitable plotting form or make a table. The coordinates of an intersection satisfy both equations. Label axes and scales, plot at least two points per line, and read the pair (x,y), not just one coordinate.
Three possible relationships
Different slopes give one intersection. Equal slopes with different intercepts give parallel distinct lines and no solution. The same line expressed differently gives infinitely many points in common. A graph reading is approximate unless separately checked exactly; a crossing outside the drawn window may still exist.
Sources for this lesson
Pause and explain
What do equal slopes and different intercepts imply?
Worked example
Find the graphical intersection of y = 2x+1 and y = −x+7, then verify it.
Show the worked solution
- The lines intersect at about (2,5).
- Equating gives 3x = 6, so x = 2; both rules give y = 5.
Answer Graph estimate (2,5), verified exactly.
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.
Sketch y = x+1 and y = −x+5. Read and check the intersection.
Give me a hint
Use two points for each line.
Compare my reasoning
- They meet at (2,3).
- 2+1 = 3 and −2+5 = 3.
(2,3), checked exactly.
Look for these in your work
- Give both coordinates
- Verify both rules
Compare y = 3x+2 and y = 3x−4. How many simultaneous solutions?
Give me a hint
Compare slopes and intercepts.
Compare my reasoning
- Both slopes are 3; intercepts differ.
- The lines are parallel and distinct.
No solution.
Look for these in your work
- Separate equal slopes from identical lines
- Explain the graph relationship
Compare 2x+y = 6 with 4x+2y = 12. What would their graphs and solution set show?
Give me a hint
Divide the second equation by 2.
Compare my reasoning
- The equations become identical.
- Both graphs are y = 6−2x; all points on that line satisfy both.
Coincident lines; infinitely many solutions (x, 6−2x).
Look for these in your work
- Recognise an equivalent equation
- Describe the whole solution set
Common mistakes
- A crossing outside the window is not evidence of no solution.
- A solution is a coordinate pair.
What can you explain now?
What do equal slopes and different intercepts imply?
Compare with the explanation
Parallel distinct lines, so no simultaneous solution.
They never have equal outputs at the same input.
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Make it stick
What do equal slopes and different intercepts imply?
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.