Final answer:
The equation of the plane that passes through the points (1, 5, 1) and is perpendicular to 2x + y - 2z = 2 and x + 3z = 4 is 2x + y - 2z - 5 = 0.
Step-by-step explanation:
To find the equation of the plane, we need to find the normal vector of the plane. The normal vector is perpendicular to the plane. In this case, we have two planes, so we can find their normal vectors by looking at the coefficients of x, y, and z. For the first plane, 2x + y - 2z = 2, the normal vector is (2, 1, -2). For the second plane, x + 3z = 4, the normal vector is (1, 0, 3).
Since the plane we want passes through the point (1, 5, 1), we can substitute the coordinates into the equation of the plane to find the constant term. Using the point (1, 5, 1) and the normal vector (2, 1, -2), we can substitute these values into the equation ax + by + cz + d = 0, and solve for d:
2(1) + 1(5) - 2(1) + d = 0
2 + 5 - 2 + d = 0
d = -5
Therefore, the equation of the plane that passes through the points (1, 5, 1) and is perpendicular to the planes 2x + y - 2z = 2 and x + 3z = 4 is 2x + y - 2z - 5 = 0.