for what values of k the system has infinite solution`{:(8x + 5y = 9),(kx + 10y = 18):}`

Answer

To find the value of \( k \) for which the system of equations has infinite solutions, we start with the given equations:

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

### Step 1: Identify the condition for infinite solutions
For a system of linear equations to have infinite solutions, the ratios of the coefficients of \( x \), \( y \), and the constant terms must be equal. This can be expressed as:

\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

Where:
- \( a_1 = 8 \), \( b_1 = 5 \), \( c_1 = -9 \) (from Equation 1)
- \( a_2 = k \), \( b_2 = 10 \), \( c_2 = -18 \) (from Equation 2)

### Step 2: Set up the ratios
From the above, we can set up the following ratios:

\[ \frac{8}{k} = \frac{5}{10} = \frac{-9}{-18} \]

### Step 3: Simplify the ratios
First, simplify \( \frac{5}{10} \) and \( \frac{-9}{-18} \):

\[ \frac{5}{10} = \frac{1}{2} \]
\[ \frac{-9}{-18} = \frac{1}{2} \]

This means we can equate:

\[ \frac{8}{k} = \frac{1}{2} \]

### Step 4: Cross-multiply to solve for \( k \)
Cross-multiplying gives us:

\[ 8 \cdot 2 = 1 \cdot k \]
\[ 16 = k \]

### Conclusion
Thus, the value of \( k \) for which the system of equations has infinite solutions is:

\[ \boxed{16} \]