The line `(x-4)/(1) =(2y-4)/(2)=(k-z)/(-2)` lies exaclty in the plane 2x-4y+z=7 find the value of k

Answer

To find the value of \( k \) such that the line given by the equation

\[ \frac{x-4}{1} = \frac{2y-4}{2} = \frac{k-z}{-2} \]

lies exactly in the plane defined by the equation

\[ 2x - 4y + z = 7, \]

we can follow these steps:

### Step 1: Express \( x \), \( y \), and \( z \) in terms of a parameter \( \lambda \)

From the line equation, we can express \( x \), \( y \), and \( z \) as follows:

1. From \( \frac{x-4}{1} = \lambda \):
\[ x = \lambda + 4 \]

2. From \( \frac{2y-4}{2} = \lambda \):
\[ 2y - 4 = 2\lambda \implies 2y = 2\lambda + 4 \implies y = \lambda + 2 \]

3. From \( \frac{k-z}{-2} = \lambda \):
\[ k - z = -2\lambda \implies z = k + 2\lambda \]

### Step 2: Substitute \( x \), \( y \), and \( z \) into the plane equation

Now we substitute \( x \), \( y \), and \( z \) into the plane equation \( 2x - 4y + z = 7 \):

Substituting the values:
\[ 2(\lambda + 4) - 4(\lambda + 2) + (k + 2\lambda) = 7 \]

### Step 3: Simplify the equation

Now we simplify the left-hand side:
\[ 2\lambda + 8 - 4\lambda - 8 + k + 2\lambda = 7 \]

Combining like terms:
\[ (2\lambda - 4\lambda + 2\lambda) + (8 - 8) + k = 7 \]
\[ 0 + k = 7 \]

### Step 4: Solve for \( k \)

From the equation, we find:
\[ k = 7 \]

Thus, the value of \( k \) is \( 7 \).