Question

If the equations \(x + 2y = 5\) and \(3x + ky = 10\) are inconsistent, then the value of \(k\) is:

• A. 4
• B. 6
• C. 8
• D. 10

Hint:

For a system of linear equations to be inconsistent, the lines must be parallel but not coincident. This is true when the ratios of the coefficients of \(x\) and \(y\) are equal, but the constant terms are not proportional.

Answer 1

The Correct Option is B

For the system of equations to be inconsistent, the lines represented by the equations must be parallel. The condition for two lines to be parallel is that their coefficients of \(x\) and \(y\) should be proportional, but the constant terms should not be proportional. The given equations are: \[ x + 2y = 5 \tag{1} \] \[ 3x + ky = 10 \tag{2} \] Step 1: Write both equations in the general form \(Ax + By = C\). Equation (1): \(x + 2y = 5\), so the coefficients of \(x\), \(y\), and the constant are \(A = 1\), \(B = 2\), and \(C = 5\). Equation (2): \(3x + ky = 10\), so the coefficients of \(x\), \(y\), and the constant are \(A = 3\), \(B = k\), and \(C = 10\). Step 2: For the lines to be parallel, the ratios of the coefficients of \(x\) and \(y\) should be equal: \[ \frac{1}{3} = \frac{2}{k} \] Step 3: Solve for \(k\): \[ k = 6 \] Thus, the value of \(k\) is 6.