If the lattice point `P(x, y, z) , x, y, zgto and x, y, zinI` with least value of z such that the 'p' lies on the planes `7x+6y+2z=272 and x-y+z=16,` then the value of `(x+y+z-42)` is equal to

Answer

To solve the problem, we need to find the lattice point \( P(x, y, z) \) such that \( x, y, z > 0 \) and \( P \) lies on the planes given by the equations \( 7x + 6y + 2z = 272 \) and \( x - y + z = 16 \). We want to minimize the value of \( z \) and then compute \( x + y + z - 42 \).

### Step 1: Express one variable in terms of the others
From the second equation \( x - y + z = 16 \), we can express \( y \) in terms of \( x \) and \( z \):
\[ y = x + z - 16 \quad \text{(Equation 1)} \]

### Step 2: Substitute \( y \) into the first equation
Now, substitute \( y \) from Equation 1 into the first equation \( 7x + 6y + 2z = 272 \):
\[ 7x + 6(x + z - 16) + 2z = 272 \]
Expanding this gives:
\[ 7x + 6x + 6z - 96 + 2z = 272 \]
Combining like terms:
\[ 13x + 8z - 96 = 272 \]

### Step 3: Rearranging the equation
Now, rearranging the equation to isolate \( z \):
\[ 13x + 8z = 272 + 96 \]
\[ 13x + 8z = 368 \]
\[ 8z = 368 - 13x \]
\[ z = \frac{368 - 13x}{8} \quad \text{(Equation 2)} \]

### Step 4: Find integer values for \( x \) and \( z \)
To minimize \( z \), we need \( 368 - 13x \) to be non-negative and divisible by 8. Thus, we need to find suitable values for \( x \).

### Step 5: Check values of \( x \)
Let’s check values of \( x \):
- For \( x = 24 \):
\[ z = \frac{368 - 13 \times 24}{8} = \frac{368 - 312}{8} = \frac{56}{8} = 7 \]
Now substituting \( x = 24 \) into Equation 1 to find \( y \):
\[ y = 24 + 7 - 16 = 15 \]

### Step 6: Verify the values
Now we have \( x = 24 \), \( y = 15 \), and \( z = 7 \). We need to check if these values satisfy both original equations:
1. For \( 7x + 6y + 2z = 272 \):
\[ 7(24) + 6(15) + 2(7) = 168 + 90 + 14 = 272 \quad \text{(True)} \]
2. For \( x - y + z = 16 \):
\[ 24 - 15 + 7 = 16 \quad \text{(True)} \]

### Step 7: Calculate \( x + y + z - 42 \)
Now we calculate:
\[ x + y + z - 42 = 24 + 15 + 7 - 42 = 46 - 42 = 4 \]

Thus, the final answer is:
\[ \boxed{4} \]