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):
\[ 3x – 4y – 1 = 0 \]
\[ 2x – \frac{8}{3}y + 5 = 0 \]
Solution
Step 1: Convert Both Equations into Comparable Form
Equation (1):
\[ 3x – 4y – 1 = 0 \Rightarrow 3x – 4y = 1 \]
Equation (2):
\[ 2x – \frac{8}{3}y + 5 = 0 \]
Multiply throughout by 3 to remove the fraction:
\[ 6x – 8y + 15 = 0 \Rightarrow 6x – 8y = -15 \]
Step 2: Compare the Two Equations
Rewrite Equation (1) by multiplying by 2:
\[ 6x – 8y = 2 \]
Thus, the equations become:
\[ 6x – 8y = 2 \quad \text{and} \quad 6x – 8y = -15 \]
They have the same coefficients of \(x\) and \(y\) but different constant terms.
Hence, the two straight lines are parallel.
Step 3: Prepare Tables of Values
For Equation (1): \(3x – 4y = 1\)
| x | y |
|---|---|
| 1 | \(\frac{1}{2}\) |
| 5 | \(\frac{7}{2}\) |
For Equation (2): \(6x – 8y = -15\)
| x | y |
|---|---|
| 1 | \(\frac{21}{8}\) |
| 5 | \(\frac{45}{8}\) |
Step 4: Graphical Representation
Plot the points:
- Line 1: \((1, \tfrac{1}{2})\) and \((5, \tfrac{7}{2})\)
- Line 2: \((1, \tfrac{21}{8})\) and \((5, \tfrac{45}{8})\)
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.