`{:(8x + 5y = 9),(kx + 10y = 18):}`

Answer

To find the value of \( k \) for which the given system of equations has infinitely many solutions, we will use the condition that the ratios of the coefficients of \( x \), \( y \), and the constant terms must be equal.

The given equations are:

1. \( 8x + 5y = 9 \) (Equation 1)
2. \( kx + 10y = 18 \) (Equation 2)

### Step 1: Write the equations in the standard form
The equations are already in the standard form \( Ax + By = C \).

### Step 2: Identify coefficients
From Equation 1:
- \( A_1 = 8 \)
- \( B_1 = 5 \)
- \( C_1 = 9 \)

From Equation 2:
- \( A_2 = k \)
- \( B_2 = 10 \)
- \( C_2 = 18 \)

### Step 3: Set up the ratio condition for infinite solutions
For the system to have infinitely many solutions, the following condition must hold:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} \]

### Step 4: Write the ratios
Substituting the values we identified:
\[ \frac{8}{k} = \frac{5}{10} = \frac{9}{18} \]

### Step 5: Simplify the ratios
The ratio \( \frac{5}{10} \) simplifies to \( \frac{1}{2} \) and \( \frac{9}{18} \) also simplifies to \( \frac{1}{2} \). Thus, we have:
\[ \frac{8}{k} = \frac{1}{2} \]

### Step 6: Cross-multiply to solve for \( k \)
Cross-multiplying gives:
\[ 8 \cdot 2 = 1 \cdot k \]
\[ 16 = k \]

### Conclusion
Thus, the value of \( k \) for which the system of equations has infinitely many solutions is:
\[ \boxed{16} \]