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):
\[ 3x – 5y = 20 \]
\[ 6x – 10y = -40 \]
Solution
Step 1: Write Both Equations
Equation (1):
\[ 3x – 5y = 20 \]
Equation (2):
\[ 6x – 10y = -40 \]
Step 2: Compare the Two Equations
Divide Equation (2) by 2:
\[ \frac{6x – 10y = -40}{2} \Rightarrow 3x – 5y = -20 \]
Thus, the equations become:
\[ 3x – 5y = 20 \quad \text{and} \quad 3x – 5y = -20 \]
They have the same coefficients of \(x\) and \(y\), but different constants.
Step 3: Prepare Tables of Values
For Equation (1): \(3x – 5y = 20\)
| x | y |
|---|---|
| 0 | -4 |
| 5 | -1 |
For Equation (2): \(6x – 10y = -40\)
(or \(3x – 5y = -20\))
| x | y |
|---|---|
| 0 | 4 |
| 5 | 7 |
Step 4: Graphical Representation
Plot the points:
- Line 1: (0, −4) and (5, −1)
- Line 2: (0, 4) and (5, 7)
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.