Final answer:
To find the parametric equations of a line parallel to another and passing through a specific point, use the direction vector from the parallel line and the given point as the line's position vector. Then apply the vector form of a line equation to find the parametric equations.
Step-by-step explanation:
To find the parametric equations for the line that passes through the point (-8, 6, 7) and is parallel to the line with parametric equations x = 1/2t, y = 1/3t, z = t, we need to use the direction vector given by the coefficients of t in the parallel line equations, which is (1/2, 1/3, 1). Then, the parametric equations will have the form x(t) = x₀ + at, y(t) = y₀ + bt, z(t) = z₀ + ct where (x₀, y₀, z₀) is the point the line passes through and (a, b, c) is the direction vector. Therefore, the parametric equations are:
x(t) = -8 + (1/2)t
y(t) = 6 + (1/3)t
z(t) = 7 + t