If the graph of given linear equations `3x+ky-4=0` and `k-4y-3x=0` coincides with each other, then the value of k is

Options

A. 3
B. 4
C. `-3`
D. `-4`

Answer

To find the value of \( k \) such that the graphs of the equations \( 3x + ky - 4 = 0 \) and \( k - 4y - 3x = 0 \) coincide, we can follow these steps:

### Step 1: Rewrite the second equation
The second equation can be rearranged to the standard form:
\[ k - 4y - 3x = 0 \implies 3x + 4y - k = 0 \]

### Step 2: Identify coefficients
Now we have two equations:
1. \( 3x + ky - 4 = 0 \) (let's call this Equation 1)
2. \( 3x + 4y - k = 0 \) (let's call this Equation 2)

From these equations, we can identify the coefficients:
- For Equation 1: \( a_1 = 3, b_1 = k, c_1 = -4 \)
- For Equation 2: \( a_2 = 3, b_2 = 4, c_2 = -k \)

### Step 3: Set up the condition for coinciding lines
For the two lines to coincide, the ratios of the coefficients must be equal:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]
Substituting the coefficients, we get:
\[ \frac{3}{3} = \frac{k}{4} = \frac{-4}{-k} \]

### Step 4: Simplify the ratios
From \( \frac{3}{3} = 1 \), we have:
\[ 1 = \frac{k}{4} \implies k = 4 \]
And from \( 1 = \frac{-4}{-k} \):
\[ 1 = \frac{4}{k} \implies k = 4 \]

### Step 5: Conclusion
Thus, the value of \( k \) such that the two lines coincide is:
\[ \boxed{4} \]