Consider the following pairs of linear equations in two variables.
Pair 1: 2x + y = 6 and x − y = 2
Pair 2: x + y = 4 and 2x + 2y = 8
Pair 3: 3x − 2y = 6 and 6x − 4y = 12
Pair 4: x − 2y = 4 and 2x − 4y = 6
For each pair, prepare a table of values (with at least two ordered pairs). Plot the points on the Cartesian plane. Draw the straight lines.
Answer: Pair 1: the lines intersect at (83, 23). Pairs 2 and 3: each pair is the same line (coincident). Pair 4: the lines are parallel.
Step-by-step solution
Idea: For each equation, the points where it meets the axes (x = 0 and y = 0) make an easy table. Two equations that are multiples of each other give the same points, so the same line.
- Pair 1. 2x + y = 6: (0, 6), (3, 0). x − y = 2: (0, −2), (2, 0). The lines cross; solving, 3x = 8, so they meet at (83, 23) ≈ (2.67, 0.67).1 mark
- Pair 2. x + y = 4: (0, 4), (4, 0). 2x + 2y = 8: (0, 4), (4, 0) — the very same points, so the lines coincide.1 mark
- Pair 3. 3x − 2y = 6: (0, −3), (2, 0). 6x − 4y = 12: (0, −3), (2, 0) — again the same line.1 mark
- Pair 4. x − 2y = 4: (0, −2), (4, 0). 2x − 4y = 6: (0, −1.5), (3, 0). Both have slope 12 but different intercepts, so the lines are parallel and never meet.1 mark
Pair 1 intersects at (8/3, 2/3); Pairs 2 and 3 are coincident lines; Pair 4 gives parallel lines.
Check: Pair 1: 2(83) + 23 = 183 = 6 ✓ and 83 − 23 = 2 ✓. Pair 4 ratios: 12 = −2−4 ≠ 46 ✓.
Answer to write in the exam
| Pair | Equation | Points | Lines |
|---|---|---|---|
| 1 | 2x + y = 6 / x − y = 2 | (0, 6), (3, 0) / (0, −2), (2, 0) | intersect at (83, 23) |
| 2 | x + y = 4 / 2x + 2y = 8 | (0, 4), (4, 0) / (0, 4), (4, 0) | coincident |
| 3 | 3x − 2y = 6 / 6x − 4y = 12 | (0, −3), (2, 0) / (0, −3), (2, 0) | coincident |
| 4 | x − 2y = 4 / 2x − 4y = 6 | (0, −2), (4, 0) / (0, −1.5), (3, 0) | parallel |
∴ Pair 1: unique solution; Pairs 2, 3: infinitely many; Pair 4: no solution
Common mistakes that cost marks
- Plotting (0, −2) at (−2, 0).
- Drawing Pair 4 as one line because the equations look alike; check the axis points, which differ.
- Reading the Pair 1 intersection as (3, 1) from a rough sketch instead of solving.
How this can come in the exam
MCQ (1 mark)
The lines x − 3y = 3 and 3x − 9y = 2 are
- intersecting
- coincident
- parallel
- perpendicular
Show answer
(C) parallel
13 = −3−9 ≠ 32.
Try one yourself
Draw x + y = 3 and 2x − y = 0 and find where they meet.
Show answer
Points (0, 3), (3, 0) and (0, 0), (1, 2): the lines meet at (1, 2).
More questions like this
- How do we find solutions from the graph? What can we say about the number of solutions when the two lines (i) intersect at a point, (ii) are parallel, (iii) are coincident?
- Solve the pair of linear equations x + 3y = 6 and 2x − 3y = 12 graphically. Comment on the nature of the solution.
- Determine if the lines x + 2y − 4 = 0 and 2x + 4y − 12 = 0 intersect, coincide or are parallel to each other. Then verify your answer by graphing the equations.
- Can you prove that two lines of equal slope are parallel?
(Hint: Consider the equations of the two lines to be y = mx + d1 and y = mx + d2 ) - Romila went to a stationery shop and purchased 2 erasers and 3 A4 sheets for ₹9. Her friend Sonali saw the new variety of erasers and A4 sheets Romila bought, and she also bought 4 erasers and 6 A4 sheets of the same kind for ₹18. Represent this situation graphically.
All Linear equations in two variables questions · All maths questions