Final answer:
To find the equations for the planes tangent to z = x² - 6x + y³ that are parallel to the plane 4x - 12y + z = 7, we set up a system of equations using the coefficients of the variables. Solving this system will give us the values needed to construct the tangent plane equations.
Step-by-step explanation:
To find the equations for the planes tangent to z = x² - 6x + y³ that are parallel to the plane 4x - 12y + z = 7, we need to compare the coefficients of the variables. The plane parallel to 4x - 12y + z = 7 will have the same coefficients of x, y, and z. So, we can set up the following system of equations:
x² - 6x + y³ = ax + by + cz
4x - 12y + z = ax + by + cz
Solving this system of equations will give us the values of a, b, and c, which we can use to construct the equations for the tangent planes.