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 avoid working with fractions, clear the denominators of the first equation by multiplying the entire equation by the Least Common Multiple (LCM) of 2 and 3, which is 6:
\[ 6\left(\frac{x}{2} + \frac{y}{3}\right) = 6(5) \implies 3x + 2y = 30 \] Now compare \(3x + 2y = 30\) with \(2x + ky = 7\).
The ratio of the coefficients of \(x\) and \(y\) is:
\[ \frac{3}{2} = \frac{2}{k} \implies 3k = 4 \implies k = \frac{4}{3} \] This is a much cleaner and less error-prone calculation!
abstract abstract

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 are given two linear equations and we need to find the value of the constant \(k\) that makes the system inconsistent.

Step 2: Key Formula or Approach:
For a pair of linear equations:
\[ a_1x + b_1y + c_1 = 0 \] \[ a_2x + b_2y + c_2 = 0 \] The algebraic condition for the system to be inconsistent (representing 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 \(ax + by + c = 0\):
First Equation: \(\frac{x}{2} + \frac{y}{3} - 5 = 0\)
Second Equation: \(2x + ky - 7 = 0\)

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

• Apply the condition for inconsistency:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \] Substitute the coefficient values:
\[ \frac{\frac{1}{2}}{2} = \frac{\frac{1}{3}}{k} \neq \frac{-5}{-7} \]

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

• Solve the first part of the proportion to find \(k\):
\[ \frac{1}{4} = \frac{1}{3k} \] Cross-multiply to solve:
\[ 3k = 4 \] \[ k = \frac{4}{3} \]

• Verify the inequality part of the condition:
Substitute \(k = \frac{4}{3}\) back into the ratio:
\[ \frac{1}{4} \neq \frac{5}{7} \] Since \(\frac{1}{4}\) is indeed not equal to \(\frac{5}{7}\), the condition for parallel lines is fully satisfied.


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