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: Rewrite both lines in slope form.
First line: \(\frac{x}{2} + \frac{y}{3} = 5\), multiply by 6 to get \(3x + 2y = 30\), so \(y = -\frac{3}{2}x + 15\), giving slope \(-\frac{3}{2}\). Second line: \(2x + ky = 7\), so \(y = -\frac{2}{k}x + \frac{7}{k}\), giving slope \(-\frac{2}{k}\).
Step 2: Use the condition for parallel lines.
Two lines are parallel, and hence the system is inconsistent, only when they have equal slopes but different intercepts.
Step 3: Equate the slopes and solve for k.
\[ -\frac{3}{2} = -\frac{2}{k} \implies 3k = 4 \implies k = \frac{4}{3} \]
Step 4: Check the intercepts differ.
With \(k = \frac{4}{3}\), the intercepts are \(15\) and \(\frac{7}{4/3} = \frac{21}{4}\), which are different, confirming the lines are parallel and the system is inconsistent. This matches option (B).
\[ \boxed{k = \frac{4}{3}} \]