Condition for Inconsistency of a Pair of Linear Equations

Video Explanation

Question

For what value of \(k\) will the following system of equations be inconsistent?

\[ 4x + 6y = 11, \qquad 2x + ky = 7 \]

Solution

Step 1: Write in Standard Form

\[ 4x + 6y – 11 = 0 \quad (1) \]

\[ 2x + ky – 7 = 0 \quad (2) \]

Step 2: Identify Coefficients

From equations (1) and (2),

\[ a_1 = 4, \quad b_1 = 6, \quad c_1 = -11 \]

\[ a_2 = 2, \quad b_2 = k, \quad c_2 = -7 \]

Step 3: Condition for Inconsistency

A pair of linear equations is inconsistent if

\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \]

Step 4: Apply the Condition

\[ \frac{a_1}{a_2} = \frac{4}{2} = 2 \]

So,

\[ \frac{6}{k} = 2 \]

\[ k = 3 \]

Step 5: Verify with Third Ratio

\[ \frac{c_1}{c_2} = \frac{11}{7} \neq 2 \]

Hence,

\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \]

So, the system is inconsistent.

Conclusion

The given system of equations is inconsistent for:

\[ \boxed{k = 3} \]

\[ \therefore \quad 4x + 6y = 11 \text{ and } 2x + 3y = 7 \text{ represent parallel lines.} \]