Question

write an equation for the distance the car travels, y, after x hours. y = 40x
how can you write an equation for any proportional relationship with slope m?
y = \frac{m}{x} \quad y = \frac{m}{x} \quad y = x + m \quad y = mx
graph: distance (mi) on y - axis (0,20,40,60,80), time (h) on x - axis (0,1,2,3,4), line through origin with positive slope

Explanation:

Step1: Identify the relationship type

The problem is about a proportional relationship (distance and time for a car). Proportional relationships have the form \( y = mx \), where \( m \) is the constant of proportionality (slope), and there is no \( y \)-intercept (since when \( x = 0 \), \( y = 0 \)).

Step2: Analyze the given example and general case

For the car, the example equation is \( y = 40x \), where 40 is the slope (speed). For any proportional relationship with slope \( m \), the equation should follow the form of a direct proportion, which is \( y = mx \) (since proportional relationships pass through the origin, so the \( y \)-intercept is 0, eliminating \( y = x + m \), and the other options \( y=\frac{m}{x} \) and \( y=\frac{m}{x} \) are inverse proportions, not direct like distance - time here).

Answer:

For the car's distance equation: \( y = 40x \) (where \( y \) is distance in miles, \( x \) is time in hours, and 40 is the speed in mph). For any proportional relationship with slope \( m \), the equation is \( y = mx \).