If `ax-2y-1=0` and `6x-4y+b=0` represent the same line, find the values of 'a' and 'b'`.

Answer

To find the values of 'a' and 'b' such that the lines represented by the equations \( ax - 2y - 1 = 0 \) and \( 6x - 4y + b = 0 \) are the same, we can follow these steps:

### Step 1: Write the equations in standard form
The equations of the lines are:
1. \( ax - 2y - 1 = 0 \)
2. \( 6x - 4y + b = 0 \)

### Step 2: Identify the coefficients
For two lines to be the same, the ratios of the coefficients of \( x \), \( y \), and the constant term must be equal. Therefore, we can write:
\[ \frac{a}{6} = \frac{-2}{-4} = \frac{-1}{b} \]

### Step 3: Simplify the ratios
First, simplify the second ratio:
\[ \frac{-2}{-4} = \frac{1}{2} \]
So we have:
\[ \frac{a}{6} = \frac{1}{2} \]

### Step 4: Solve for 'a'
Cross-multiply to solve for \( a \):
\[ a \cdot 2 = 6 \cdot 1 \implies 2a = 6 \implies a = \frac{6}{2} = 3 \]

### Step 5: Use the first ratio to find 'b'
Now, using the first and third ratios:
\[ \frac{1}{2} = \frac{-1}{b} \]
Cross-multiply to solve for \( b \):
\[ 1 \cdot b = -2 \cdot (-1) \implies b = -2 \]

### Final Values
Thus, we find:
\[ a = 3 \quad \text{and} \quad b = -2 \]

### Summary of the solution:
The values of \( a \) and \( b \) are:
- \( a = 3 \)
- \( b = -2 \)