Graphical Representation of an Inconsistent Pair of Linear Equations
Video Explanation
Question
Show graphically that the following system of equations is inconsistent (i.e. has no solution):
\[ 2y – x = 9 \]
\[ 6y – 3x = 21 \]
Solution
Step 1: Write Both Equations in Comparable Form
Equation (1):
\[ 2y – x = 9 \Rightarrow x = 2y – 9 \]
Equation (2):
\[ 6y – 3x = 21 \Rightarrow 2y – x = 7 \Rightarrow x = 2y – 7 \]
Step 2: Compare the Equations
The equations are:
\[ 2y – x = 9 \quad \text{and} \quad 2y – x = 7 \]
They have the same coefficients of \(x\) and \(y\) but different constant terms. Hence, the corresponding lines are parallel.
Step 3: Prepare Tables of Values
For Equation (1): \(2y – x = 9\)
| x | y |
|---|---|
| -9 | 0 |
| -5 | 2 |
For Equation (2): \(2y – x = 7\)
| x | y |
|---|---|
| -7 | 0 |
| -3 | 2 |
Step 4: Graphical Representation
Plot the points:
- Line 1: (−9, 0) and (−5, 2)
- Line 2: (−7, 0) and (−3, 2)
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 are parallel and do not intersect, the given system of equations has no solution.
Hence, the system of equations is inconsistent.