To check whether a line belongs to a plane, you can substitute the coordinates of a point on the line into the equation of the plane. If the equation is satisfied, then the point lies on the plane and the line belongs to the plane.
For example, suppose you have the Cartesian equation of a line in the form:
y = mx + b
and the equation of a plane in the form:
ax + by + cz + d = 0
To check if the line belongs to the plane, you can substitute a point (x, y) on the line into the equation of the plane and solve for z. If the resulting value of z satisfies the equation of the plane, then the line belongs to the plane.
Here's an example:
Suppose the Cartesian equation of the line is y = 2x + 1, and the equation of the plane is 2x + 3y + 4z + 5 = 0.
If we substitute a point on the line, such as (1, 3), into the equation of the plane, we get:
2(1) + 3(3) + 4z + 5 = 0
Solving for z, we find that z = -1.5.
Since the resulting value of z satisfies the equation of the plane, the line belongs to the plane.
If the resulting value of z does not satisfy the equation of the plane, then the line does not belong to the plane.