For what values of k will the given equations in two variables represent coincident lines: `2x+32y+3=0` and `3x+48y+k=0`?

Options

A. `2/3`
B. `3/2`
C. `9/2`
D. `1`

Answer

To determine the values of \( k \) for which the equations \( 2x + 32y + 3 = 0 \) and \( 3x + 48y + k = 0 \) represent coincident lines, we need to use the condition for coincident lines. The lines will be coincident if the ratios of the coefficients of \( x \), \( y \), and the constant term are equal.

### Step-by-step Solution:

1. **Identify coefficients from the equations:**
- For the first equation \( 2x + 32y + 3 = 0 \):
- \( a_1 = 2 \)
- \( b_1 = 32 \)
- \( c_1 = 3 \)
- For the second equation \( 3x + 48y + k = 0 \):
- \( a_2 = 3 \)
- \( b_2 = 48 \)
- \( c_2 = k \)

2. **Set up the ratio condition for coincident lines:**
- The lines are coincident if:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]
- This gives us the following equations:
\[ \frac{2}{3} = \frac{32}{48} = \frac{3}{k} \]

3. **Simplify the second ratio:**
- Simplifying \( \frac{32}{48} \):
\[ \frac{32}{48} = \frac{2}{3} \]
- So, we confirm that:
\[ \frac{2}{3} = \frac{2}{3} \]

4. **Set up the equation for \( k \):**
- Now we need to solve:
\[ \frac{3}{k} = \frac{2}{3} \]

5. **Cross-multiply to find \( k \):**
- Cross-multiplying gives:
\[ 2k = 9 \]
- Dividing both sides by 2:
\[ k = \frac{9}{2} = 4.5 \]

### Final Answer:
The value of \( k \) for which the given equations represent coincident lines is \( k = 4.5 \).

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