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.