For what value of k, the system of equations 8x + 5y = 9 and kx + 10y = 18 has infinitely many solutions?

Options

A. `k= 10 `
B. `k = 16`
C. `k = 8 `
D. `k = 15 `

Answer

To find the value of \( k \) for which the system of equations

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

has infinitely many solutions, we can follow these steps:

### Step 1: Understand the condition for infinitely many solutions
For two linear equations to have infinitely many solutions, they must represent the same line. This occurs when the ratios of the coefficients of \( x \), \( y \), and the constant terms are equal. Mathematically, this is expressed as:

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

where \( a_1, b_1, c_1 \) are the coefficients from the first equation and \( a_2, b_2, c_2 \) are the coefficients from the second equation.

### Step 2: Identify coefficients
From the given equations:
- For the first equation \( 8x + 5y = 9 \):
- \( a_1 = 8 \)
- \( b_1 = 5 \)
- \( c_1 = -9 \) (we rewrite it as \( 8x + 5y - 9 = 0 \))

- For the second equation \( kx + 10y = 18 \):
- \( a_2 = k \)
- \( b_2 = 10 \)
- \( c_2 = -18 \) (we rewrite it as \( kx + 10y - 18 = 0 \))

### Step 3: Set up the ratios
Now we set up the ratios:

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

### Step 4: Simplify the ratios
First, simplify \( \frac{5}{10} \) and \( \frac{-9}{-18} \):

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

Thus, we have:

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

### Step 5: Solve for \( k \)
Cross-multiply to solve for \( k \):

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

### Conclusion
The value of \( k \) for which the system of equations has infinitely many solutions is \( k = 16 \).

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