for what values of k the system has consistent solution ` {:(2x - 4y = k),(5x - 10y = 7.5):}`

Answer

To determine the values of \( k \) for which the system of equations has consistent solutions, we start with the given equations:

1. \( 2x - 4y = k \) (Equation 1)
2. \( 5x - 10y = 7.5 \) (Equation 2)

### Step 1: Identify the coefficients
From the equations, we can identify the coefficients:
- For Equation 1: \( a_1 = 2 \), \( b_1 = -4 \), \( c_1 = k \)
- For Equation 2: \( a_2 = 5 \), \( b_2 = -10 \), \( c_2 = 7.5 \)

### Step 2: Calculate the ratios
We will calculate the ratios of the coefficients:
- \( \frac{a_1}{a_2} = \frac{2}{5} \)
- \( \frac{b_1}{b_2} = \frac{-4}{-10} = \frac{2}{5} \)
- \( \frac{c_1}{c_2} = \frac{k}{7.5} \)

### Step 3: Determine conditions for consistent solutions
For the system to have consistent solutions, we need to check two scenarios:
1. **Unique Solution**: This occurs when \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \).
2. **Infinitely Many Solutions**: This occurs when \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \).

### Step 4: Analyze the ratios
From our calculations:
- We have \( \frac{a_1}{a_2} = \frac{2}{5} \) and \( \frac{b_1}{b_2} = \frac{2}{5} \).
- Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \), we will check for infinitely many solutions.

### Step 5: Set the ratios equal
Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \), we set it equal to \( \frac{c_1}{c_2} \):
\[ \frac{2}{5} = \frac{k}{7.5} \]

### Step 6: Solve for \( k \)
To find \( k \), cross-multiply:
\[ 2 \cdot 7.5 = 5 \cdot k \]
\[ 15 = 5k \]
Now, divide both sides by 5:
\[ k = \frac{15}{5} = 3 \]

### Conclusion
Thus, the value of \( k \) for which the system has consistent solutions is:
\[ \boxed{3} \]