Question

The value of k for which the system of equations \(3x - 7y = 1\) and \(kx + 14y = 6\) is inconsistent, is

• A. --6
• B. \(\frac{2}{3}\)
• C. 6
• D. \(-\frac{3}{2}\)

Answer 1

The Correct Option is A

Given:
Two linear equations:
1) \(3x - 7y = 1\) with \(a_1 = 3, b_1 = -7, c_1 = 1\)
2) \(kx + 14y = 6\) with \(a_2 = k, b_2 = 14, c_2 = 6\)

Step 1: Condition for inconsistency
A system of two equations is inconsistent (no solution) if:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \]

Step 2: Set \(\frac{a_1}{a_2} = \frac{b_1}{b_2}\) and solve for \(k\)
\[ \frac{3}{k} = \frac{-7}{14} \] Simplify right side:
\[ \frac{3}{k} = -\frac{1}{2} \] Cross-multiply:
\[ 3 \times 2 = -1 \times k \implies 6 = -k \implies k = -6 \]

Step 3: Check if \(\frac{b_1}{b_2} \neq \frac{c_1}{c_2}\)
Calculate:
\[ \frac{b_1}{b_2} = \frac{-7}{14} = -\frac{1}{2} \] \[ \frac{c_1}{c_2} = \frac{1}{6} \] Since \(-\frac{1}{2} \neq \frac{1}{6}\), the condition for inconsistency is satisfied.

Final Answer:
\[ \boxed{k = -6} \]