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

To determine the value of \( k \) for which the system of linear equations is inconsistent, we need to analyze the equations:

1. \(\frac{x}{2} + \frac{y}{3} = 5\)

2. \(2x + ky = 7\)

For the system of equations to be inconsistent, the lines represented by these equations must be parallel, which means they have the same slope but different intercepts.

We start by converting the first equation to a standard linear form \(ax + by = c\).

Equation 1: \(\frac{x}{2} + \frac{y}{3} = 5\)

Multiply through by 6 to clear the fractions:

\(3x + 2y = 30\)  (Equation i)

Equation 2 is already in a suitable form:

\(2x + ky = 7\)  (Equation ii)

For the lines to be parallel, the ratio of the coefficients of \(x\) and \(y\) must be equal, i.e.,

\(\frac{3}{2} = \frac{2}{k}\)

Cross-multiplying gives:

\(3k = 4\)

Solving for \(k\), we get:

\(k = \frac{4}{3}\)

Thus, the value of \(k\) for which the system of equations is inconsistent is \(\frac{4}{3}\).

Therefore, the correct answer is \(\frac{4}{3}\).