Question

For what value of \( k \) has the system of linear equations \( x + 2y = 3 \) and \( 5x + ky = 15 \) infinite solutions?

• A. 5
• B. 10
• C. 6
• D. 12

Hint:

For two linear equations to have infinite solutions, their corresponding coefficients and constants must be in the same ratio.

Answer 1

The Correct Option is B

Step 1: For the system to have infinite solutions, the two equations must be dependent. This means the ratios of the coefficients of \( x \), \( y \), and the constants should be the same. The given system is: \[ x + 2y = 3 \quad \text{(1)} \] \[ 5x + ky = 15 \quad \text{(2)} \] Step 2: Find the ratio of the coefficients of \( x \), \( y \), and the constant term: \[ \frac{1}{5} = \frac{2}{k} = \frac{3}{15} \] Step 3: From the equation \( \frac{2}{k} = \frac{3}{15} \), solve for \( k \): \[ k = \frac{2 \times 15}{3} = 10 \] Thus, the value of \( k \) for which the system has infinite solutions is \( k = 10 \).