If the equations `4x+7y=10` and `10x+ky=25` represent coincident lines then find the value of k?

Answer

To find the value of \( k \) such that the equations \( 4x + 7y = 10 \) and \( 10x + ky = 25 \) represent coincident lines, we can follow these steps:

### Step 1: Understand the condition for coincident lines
Two lines are coincident if the ratios of their coefficients are equal. This means:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]
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 equations
From the first equation \( 4x + 7y = 10 \):
- \( a_1 = 4 \)
- \( b_1 = 7 \)
- \( c_1 = 10 \)

From the second equation \( 10x + ky = 25 \):
- \( a_2 = 10 \)
- \( b_2 = k \)
- \( c_2 = 25 \)

### Step 3: Set up the ratios
Using the condition for coincident lines, we can set up the following ratios:
\[ \frac{4}{10} = \frac{7}{k} = \frac{10}{25} \]

### Step 4: Simplify the ratios
First, simplify \( \frac{4}{10} \) and \( \frac{10}{25} \):
\[ \frac{4}{10} = \frac{2}{5} \]
\[ \frac{10}{25} = \frac{2}{5} \]

Thus, we have:
\[ \frac{2}{5} = \frac{7}{k} \]

### Step 5: Cross-multiply to solve for \( k \)
Cross-multiplying gives:
\[ 2k = 5 \cdot 7 \]
\[ 2k = 35 \]

### Step 6: Solve for \( k \)
Now, divide both sides by 2:
\[ k = \frac{35}{2} \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{\frac{35}{2}} \]