Graphical Test of Consistency of a Pair of Linear Equations
Video Explanation
Question
Determine graphically whether the following system of equations is consistent or inconsistent:
\[ x – 2y = 2 \]
\[ 4x – 2y = 5 \]
Solution
Step 1: Write Both Equations in Convenient Form
Equation (1):
\[ x – 2y = 2 \Rightarrow y = \frac{x – 2}{2} \]
Equation (2):
\[ 4x – 2y = 5 \Rightarrow 2y = 4x – 5 \Rightarrow y = 2x – \frac{5}{2} \]
Step 2: Prepare Tables of Values
For Equation (1): \(x – 2y = 2\)
| x | y |
|---|---|
| 2 | 0 |
| 4 | 1 |
For Equation (2): \(4x – 2y = 5\)
| x | y |
|---|---|
| 1 | \(-\frac{3}{2}\) |
| 2 | \(\frac{3}{2}\) |
Step 3: Graphical Representation
Plot the following points on the same Cartesian plane:
- Line 1: (2, 0) and (4, 1)
- Line 2: (1, −3/2) and (2, 3/2)
Join each pair of points to obtain two straight lines.
The two straight lines intersect at exactly one point.
Conclusion
Since the two lines intersect at one point, the given system of equations has a unique solution.
Hence, the system of equations is consistent.