Question

The system of equations \( 3x + 5y = 4 \); \( 6x + ky = 8 \) has infinitely many solutions if \( k \) is equal to

• A. 0
• B. 7
• C. 10
• D. 6 or 8

Hint:

To have infinitely many solutions, the system must be consistent and dependent, meaning the ratios of the coefficients must be equal.

Answer 1

The Correct Option is D

Step 1: Condition for infinitely many solutions.
For the system of equations to have infinitely many solutions, the two equations must be dependent. This happens when the ratios of the coefficients of \(x\), \(y\), and the constants are equal. For the system: \[ \frac{3}{6} = \frac{5}{k} = \frac{4}{8} \]
Step 2: Solving for \( k \).
From the first and third ratios: \[ \frac{3}{6} = \frac{4}{8} \quad \text{(True, as both are equal to } \frac{1}{2}) \] Now, solve the middle ratio: \[ \frac{5}{k} = \frac{1}{2} \] \[ k = 10 \] Thus, the system has infinitely many solutions when \( k = 10 \). The correct answer is (4) 6 or 8.