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:

To easily work with the first equation, multiply it by 6 to clear the fractions:
\[ 6\left(\frac{x}{2} + \frac{y}{3}\right) = 6(5) \implies 3x + 2y = 30 \] Now compare coefficients with \(2x + ky = 7\):
\[ \frac{3}{2} = \frac{2}{k} \implies 3k = 4 \implies k = \frac{4}{3} \] This reduces fraction handling and saves time!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The topic of this question is a Pair of Linear Equations in Two Variables.
A system of linear equations can be consistent (having one unique solution or infinitely many solutions) or inconsistent (having no solution).
Graphically, an inconsistent system represents two parallel lines that never intersect.
We need to find the value of the constant \(k\) that makes the given system inconsistent.

Step 2: Key Formula or Approach:
For a system of linear equations:
\[ a_1x + b_1y + c_1 = 0 \] \[ a_2x + b_2y + c_2 = 0 \] The algebraic condition for inconsistency (parallel lines) is:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \] We will identify the coefficients, set up the ratio equality, and solve for \(k\).

Step 3: Detailed Explanation:

• Write the given equations in standard form:
- First Equation: \(\frac{x}{2} + \frac{y}{3} - 5 = 0\)
- Second Equation: \(2x + ky - 7 = 0\)

• Identify the coefficients from both equations:
\(a_1 = \frac{1}{2}\), \(b_1 = \frac{1}{3}\), \(c_1 = -5\)
\(a_2 = 2\), \(b_2 = k\), \(c_2 = -7\)

• Substitute these coefficients into the inconsistency ratio:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \] \[ \frac{\frac{1}{2}}{2} = \frac{\frac{1}{3}}{k} \neq \frac{-5}{-7} \]

• Simplify the fractions:
\[ \frac{1}{4} = \frac{1}{3k} \neq \frac{5}{7} \]

• Solve for \(k\) using the first ratio equality:
\[ \frac{1}{4} = \frac{1}{3k} \] Cross-multiply:
\[ 3k = 4 \implies k = \frac{4}{3} \]

• Verify the inequality:
Since \(\frac{1}{4} \neq \frac{5}{7}\), our calculated value of \(k\) satisfies the condition for inconsistency.


Step 4: Final Answer:
The system of equations is inconsistent when \(k = \frac{4}{3}\).
Therefore, the correct option is (B).