Question

1. a participant in a 21-mile walkathon walks at a steady rate of 3 miles per hour. he thinks, “the relationship between the number of miles left to walk and the number of hours i already walked can be represented by a line with slope -3.” do you agree with his claim? explain your reasoning.

Explanation:

Step1: Define variables

Let \( x \) be the number of hours walked, and \( y \) be the number of miles left to walk. The total distance of the walkathon is 21 miles, and the walking rate is 3 miles per hour. So the distance walked after \( x \) hours is \( 3x \) miles.

Step2: Formulate the equation

The number of miles left \( y \) is the total distance minus the distance walked, so \( y = 21 - 3x \).

Step3: Analyze the slope

The equation \( y = - 3x + 21 \) is in the slope - intercept form \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. Here, \( m=-3 \). This means that for each additional hour walked (\( x \) increases by 1), the number of miles left to walk (\( y \)) decreases by 3, which is consistent with the rate of walking (3 miles per hour).

Answer:

Yes, I agree with his claim. The equation relating the number of miles left (\( y \)) and hours walked (\( x \)) is \( y = 21-3x \) (or \( y=-3x + 21 \)), which is in slope - intercept form \( y = mx + b \) with \( m=-3 \), so the slope of the line representing this relationship is - 3.