To find the equation of the plane passing through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 7, and perpendicular to the plane x = 0, we can follow these steps:

1) Find the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 7.
To do this, we can use the method of solving a system of equations:

x + y + z = 1
2x + 3y + 4z = 7

Multiplying the first equation by 2, we get:
2x + 2y + 2z = 2

Subtracting this from the second equation, we get:
2x + 3y + 4z - (2x + 2y + 2z) = 7 - 2
y + 2z = 5

Now, we can set z = t (a parameter) and solve for y in terms of t:
y + 2t = 5
y = 5 - 2t

Finally, we can substitute y = 5 - 2t and z = t into the first equation to solve for x:
x + (5 - 2t) + t = 1
x + 5 - t = 1
x = -4 + t

So, the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 7 can be parameterized as:
x = -4 + t
y = 5 - 2t
z = t

2) Find a vector that is perpendicular to the plane x = 0.
Since x = 0, the plane is parallel to the yz-plane. Therefore, any vector with non-zero y and z components will be perpendicular to the plane. For example, the vector v = (0, 1, 0) is perpendicular to the plane x = 0.

3) Use the direction vector of the line of intersection and the perpendicular vector to find the normal vector of the desired plane.
The direction vector of the line of intersection is given by the coefficients of the parameters t in the equations for x, y, and z. In this case, the direction vector is d = (1, -2, 1).

To find the normal vector of the plane passing through the line of intersection and perpendicular to the plane x = 0, we can take the cross product of the direction vector d and the perpendicular vector v:
n = d × v

Using the cross product formula, we get:
n = (1, -2, 1) × (0, 1, 0)
= (1*0 - (-2*1), -(1*0 - 1*0), 1*1 - (-2*0))
= (2, 0, 1)

So, the normal vector of the desired plane is n = (2, 0, 1).

4) Write the equation of the plane using the normal vector and a point on the plane.
Since the plane passes through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z =