For what value of k, the equations 3x - y = 8 and 9x - ky = 24 will have infinitely many solutions ?
(a)6
(b)5
(c)3
(d)1

Options

A. 6
B. 5
C. 3
D. 1

Answer

To determine the value of \( k \) for which the equations \( 3x - y = 8 \) and \( 9x - ky = 24 \) have infinitely many solutions, we can follow these steps:

### Step 1: Write the equations in standard form
The given equations are:
1. \( 3x - y = 8 \)
2. \( 9x - ky = 24 \)

We can rewrite these equations in the form \( Ax + By + C = 0 \):
1. \( 3x - y - 8 = 0 \) (Here, \( A_1 = 3, B_1 = -1, C_1 = -8 \))
2. \( 9x - ky - 24 = 0 \) (Here, \( A_2 = 9, B_2 = -k, C_2 = -24 \))

### Step 2: Apply the condition for infinitely many solutions
For the two equations to have infinitely many solutions, the ratios of the coefficients must be equal:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} \]

### Step 3: Set up the equations using the ratios
Using the first two ratios:
\[ \frac{3}{9} = \frac{-1}{-k} \]

### Step 4: Simplify the ratios
Simplifying the left side:
\[ \frac{3}{9} = \frac{1}{3} \]
So we have:
\[ \frac{1}{3} = \frac{1}{k} \]

### Step 5: Cross-multiply to solve for \( k \)
Cross-multiplying gives:
\[ 1 \cdot k = 3 \cdot 1 \]
Thus:
\[ k = 3 \]

### Conclusion
The value of \( k \) for which the equations have infinitely many solutions is \( \boxed{3} \).

---