Final answer:
To find a parallel equation to the line h(t) = (1/3)x + 2, the slope remains the same, thus a possible equation is h(t) = (1/3)x + 1. A perpendicular equation that passes through the origin has the slope which is the negative reciprocal of the original slope, giving us h(t) = -3x.
Step-by-step explanation:
The original equation provided: h(t) = (1/3)x + 2, represents a linear equation in the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. To find a parallel equation to the line, we require an equation with the same slope (1/3) since parallel lines have identical slopes. The parallel line has to have a different y-intercept; however, since we're not given a specific point that the new line must pass through (other than not being the same line), any value of b except 2 will work. A suitable parallel line could be h(t) = (1/3)x + 1.
For a perpendicular equation that passes through the origin (0,0), we use the negative reciprocal of the original slope (−3). The perpendicular line will have the form y = mx + b, where m is −3 and b is 0, because it passes through the origin. Therefore, a perpendicular line through the origin is h(t) = -3x.