The value of K for which system of equation `5x + 2y = k and 10 x + 4y =30` has infintely many solution is

Options

A. `1/2`
B. 15
C. 6
D. `1/6`

Answer

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

\[ 5x + 2y = k \]
\[ 10x + 4y = 30 \]

has infinitely many solutions, we can follow these steps:

### Step 1: Identify the coefficients
The first equation is \( 5x + 2y = k \). The coefficients are:
- \( a_1 = 5 \)
- \( b_1 = 2 \)
- \( c_1 = k \)

The second equation is \( 10x + 4y = 30 \). The coefficients are:
- \( a_2 = 10 \)
- \( b_2 = 4 \)
- \( c_2 = 30 \)

### Step 2: Set up the condition for infinitely many solutions
For the system of equations 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 3: Substitute the coefficients into the condition
Substituting the coefficients into the condition, we have:
\[ \frac{5}{10} = \frac{2}{4} = \frac{k}{30} \]

### Step 4: Simplify the fractions
Now, simplify the fractions:
\[ \frac{5}{10} = \frac{1}{2} \]
\[ \frac{2}{4} = \frac{1}{2} \]

Thus, we have:
\[ \frac{1}{2} = \frac{k}{30} \]

### Step 5: Solve for \( k \)
To find \( k \), we can cross-multiply:
\[ 1 \cdot 30 = 2 \cdot k \]
\[ 30 = 2k \]

Now, divide both sides by 2:
\[ k = \frac{30}{2} = 15 \]

### Conclusion
The value of \( k \) for which the system of equations has infinitely many solutions is:
\[ \boxed{15} \]

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