Graphical Representation of an Inconsistent System of Linear Equations
Video Explanation
Question
Show graphically that the following system of equations is inconsistent (i.e. has no solution):
\[ x – 2y = 6 \]
\[ 3x – 6y = 0 \]
Solution
Step 1: Write Both Equations
Equation (1):
\[ x – 2y = 6 \]
Equation (2):
\[ 3x – 6y = 0 \Rightarrow x – 2y = 0 \]
Step 2: Compare the Equations
The equations are:
\[ x – 2y = 6 \quad \text{and} \quad x – 2y = 0 \]
They have the same coefficients of \(x\) and \(y\), but different constant terms.
Hence, the two lines are parallel.
Step 3: Prepare Tables of Values
For Equation (1): \(x – 2y = 6\)
| x | y |
|---|---|
| 6 | 0 |
| 2 | -2 |
For Equation (2): \(x – 2y = 0\)
| x | y |
|---|---|
| 0 | 0 |
| 2 | 1 |
Step 4: Graphical Representation
Plot the points:
- Line 1: (6, 0) and (2, −2)
- Line 2: (0, 0) and (2, 1)
Join each pair of points to obtain two straight lines.
The two lines are parallel and do not intersect.
Conclusion
Since the two straight lines do not intersect, the given system of equations has no solution.
Hence, the system of equations is inconsistent.