Question
Solve the following pair of linear equations graphically:
Find the point of intersection and verify your answer algebraically.
(NCERT Class 10, Chapter 3 — Pair of Linear Equations in Two Variables)
Solution — Step by Step
We need at least two points per line to plot them. First, rearrange:
Line 1:
Line 2:
Pick convenient values of :
| Point | ||
|---|---|---|
| 0 | 14 | (0, 14) |
| 7 | 7 | (7, 7) |
| 14 | 0 | (14, 0) |
Plot these three points and draw a straight line through them.
| Point | ||
|---|---|---|
| 0 | -4 | (0, -4) |
| 4 | 0 | (4, 0) |
| 7 | 3 | (7, 3) |
Plot these points and draw the second line.
Both lines pass through the point (9, 5). This is the solution.
Let’s verify: (checks out) and (checks out).
So , .
Why This Works
When we plot two linear equations on the same axes, their point of intersection gives the values of and that satisfy both equations simultaneously. That’s exactly what “solving a pair of linear equations” means — finding the common solution.
If the lines are parallel (same slope, different intercepts), there’s no solution. If they overlap completely, there are infinitely many solutions. Here, the slopes are different ( and ), so we get exactly one intersection point.
Alternative Method — Elimination
Add the two equations directly:
Substitute back: .
For the graphical method in board exams, always use a scale of at least 1 cm = 1 unit. CBSE marking scheme gives marks for: correct table of values (1 mark), correct plotting of both lines (2 marks), and correctly reading the intersection point (1 mark). Don’t skip the table — even if you know the answer.
Common Mistake
Students often plot only two points per line and then draw a slightly inaccurate line — which makes the intersection appear at the wrong coordinate. Always plot at least three points per line. The third point acts as a self-check: if all three don’t lie on a straight line, you’ve made an arithmetic error in your table.