Question

car x travels 144 miles in 3 hours.
a. write the equation of the line that describes the relationship between distance and time. use x for the time in hours and y for the distance in miles.
b. what is the graph that represents the relationship between distance and time for car x? explain.
if a car travels for 0 hours, it will travel \\(\square\\) mile(s). this means it will pass through the point \\(\square\\). use the slope to move 3 units to the right of the origin and \\(\square\\) unit(s) up to find the point that can be used to graph the relationship.

Explanation:

Part a: Equation of the Line

Step 1: Identify the relationship type

The relationship between distance \( y \) (in miles) and time \( x \) (in hours) for a car is typically linear (constant speed), so we use the slope - intercept form \( y = mx + b \). When \( x = 0 \) (time = 0 hours), the distance \( y = 0 \) (since the car starts from rest), so \( b = 0 \).

Step 2: Calculate the slope (speed)

The car travels 144 miles in 3 hours. The slope \( m \) (which represents speed, distance per unit time) is calculated as \( m=\frac{\text{change in }y}{\text{change in }x}=\frac{144}{3} = 48 \).

Step 3: Write the equation

Since \( b = 0 \) and \( m = 48 \), the equation of the line is \( y=48x \).

The graph of \( y = 48x \) is a straight line passing through the origin \((0,0)\) with a slope of 48. In the context of distance - time for Car X, the \( x \) - axis represents time (in hours) and the \( y \) - axis represents distance (in miles). The slope of 48 means that for every 1 - hour increase in time (\( x \) - value), the distance (\( y \) - value) increases by 48 miles. This shows a proportional relationship between time and distance, indicating that Car X is moving at a constant speed of 48 miles per hour. As time (\( x \)) increases, distance (\( y \)) increases linearly according to the equation \( y = 48x \).

Answer:

The equation of the line is \( \boldsymbol{y = 48x} \)

Part b: Interpretation of the Graph