`{:(2x - 4y = 3),(5x - ky = 7.5):}`

Answer

To solve the given system of equations \(2x - 4y = 3\) and \(5x - ky = 7.5\), we will analyze the conditions for unique solutions and infinite solutions based on the values of \(k\).

### Step 1: Rewrite the equations in standard form
We can rewrite the equations in the standard form \(Ax + By + C = 0\).

1. For the first equation:
\[ 2x - 4y - 3 = 0 \quad \text{(Equation 1)} \]

2. For the second equation:
\[ 5x - ky - 7.5 = 0 \quad \text{(Equation 2)} \]

### Step 2: Identify coefficients
From the equations, we identify the coefficients:
- For Equation 1: \(A_1 = 2\), \(B_1 = -4\), \(C_1 = -3\)
- For Equation 2: \(A_2 = 5\), \(B_2 = -k\), \(C_2 = -7.5\)

### Step 3: Determine the conditions for unique and infinite solutions
To find the conditions for unique and infinite solutions, we will use the following ratios:
- Unique solution: \(\frac{A_1}{A_2} \neq \frac{B_1}{B_2}\)
- Infinite solutions: \(\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2}\)

### Step 4: Calculate the ratios
1. Calculate \(\frac{A_1}{A_2}\):
\[ \frac{A_1}{A_2} = \frac{2}{5} \]

2. Calculate \(\frac{B_1}{B_2}\):
\[ \frac{B_1}{B_2} = \frac{-4}{-k} = \frac{4}{k} \]

3. Calculate \(\frac{C_1}{C_2}\):
To avoid decimals, convert \(7.5\) to a fraction:
\[ \frac{C_1}{C_2} = \frac{-3}{-7.5} = \frac{3}{7.5} = \frac{3 \times 2}{7.5 \times 2} = \frac{6}{15} = \frac{2}{5} \]

### Step 5: Set up the equations for unique solution
For a unique solution, we set:
\[ \frac{2}{5} \neq \frac{4}{k} \]
Cross-multiplying gives:
\[ 2k \neq 20 \quad \Rightarrow \quad k \neq 10 \]

### Step 6: Set up the equations for infinite solutions
For infinite solutions, we set:
\[ \frac{2}{5} = \frac{4}{k} \]
Cross-multiplying gives:
\[ 2k = 20 \quad \Rightarrow \quad k = 10 \]

### Conclusion
The value of \(k\) can be any real number except \(10\) for a unique solution, or \(k = 10\) for infinite solutions.