Question

yochanan walked from home to the bus stop at an average speed of 5 mph. he immediately got on his school bus and traveled at an average speed of 60 mph until he got to school. the total distance from his home to school is 35 miles, and the entire trip took 1.5 hours. the following system of equations represents this: 5x + 60y = 35, x + y = 1.5. yochanan walked how many miles?

Explanation:

Step1: Define Variables

Let \( x \) be the time (in hours) Yochanan walked, and \( y \) be the time (in hours) he rode the bus.

Step2: Analyze the First Equation

The total distance is 35 miles. Distance = speed × time. Walking speed is 5 mph, so walking distance is \( 5x \). Bus speed is 60 mph, so bus distance is \( 60y \). Thus, \( 5x + 60y = 35 \).

Step3: Analyze the Second Equation

The total time is 1.5 hours. So, \( x + y = 1.5 \).

Step4: Solve the System of Equations

From \( x + y = 1.5 \), we get \( x = 1.5 - y \). Substitute into \( 5x + 60y = 35 \):

\( 5(1.5 - y) + 60y = 35 \)

\( 7.5 - 5y + 60y = 35 \)

\( 55y = 35 - 7.5 \)

\( 55y = 27.5 \)

\( y = \frac{27.5}{55} = 0.5 \)

Then \( x = 1.5 - 0.5 = 1 \) hour.

Step5: Find Walking Distance

Walking distance = \( 5x = 5×1 = 5 \) miles.

Answer:

Yochanan walked 5 miles.