Step 1: Understanding the Question:
We use the concept of a family of planes passing through the line of intersection of two given planes.
Step 2: Key Formula or Approach:
The equation of the family is $P_1 + \lambda P_2 = 0$. We find $\lambda$ using the given point.
Step 3: Detailed Explanation:
The equation of the plane is $(x + 2y - z + 1) + \lambda(3x - y - 4z + 3) = 0$.
Since it passes through $(1, 1, 1)$, substitute these coordinates:
$(1 + 2(1) - 1 + 1) + \lambda(3(1) - 1 - 4(1) + 3) = 0$
$(3) + \lambda(1) = 0 \implies \lambda = -3$.
Substitute $\lambda = -3$ back into the family equation:
$(x + 2y - z + 1) - 3(3x - y - 4z + 3) = 0$
$x + 2y - z + 1 - 9x + 3y + 12z - 9 = 0$
$-8x + 5y + 11z - 8 = 0$
Multiplying by $-1$:
$8x - 5y - 11z + 8 = 0$.
Step 4: Final Answer:
The equation is $8x - 5y - 11z + 8 = 0$.