Question

The value of \(k\) for which the system of linear equations \(\frac{x}{2} + \frac{y}{3} = 5\) and \(2x + ky = 7\) is inconsistent, is

• A. \(\frac{3}{4}\)
• B. \(\frac{4}{3}\)
• C. \(\frac{1}{3}\)
• D. \(3\)

Hint:

Always multiply equations with fractions by their LCM to get standard forms before comparing coefficients. This prevents calculation errors.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A system of linear equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) is inconsistent (no solution) if the lines are parallel.
Step 2: Key Formula or Approach:
Condition for inconsistency: \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\).
Eq 1: \(\frac{1}{2}x + \frac{1}{3}y - 5 = 0 \Rightarrow 3x + 2y - 30 = 0\) (multiplying by 6).
Eq 2: \(2x + ky - 7 = 0\).
Step 3: Detailed Explanation:
Comparing coefficients:
\(a_1 = 3, b_1 = 2, c_1 = -30\)
\(a_2 = 2, b_2 = k, c_2 = -7\)
Apply the first part of the condition:
\[ \frac{3}{2} = \frac{2}{k} \]
Cross-multiplying:
\[ 3k = 4 \]
\[ k = \frac{4}{3} \]
Check the second part: \(\frac{3}{2} \neq \frac{-30}{-7}\), which is \(1.5 \neq 4.28\). The condition holds.
Step 4: Final Answer:
The value of \(k\) is \(\frac{4}{3}\).