Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

Question

Darren and Preston started out at their houses and are biking towards each other. Darren started out first, and has already gone 2 miles. He bikes at a constant speed of 3 miles per hour. Preston just left, and rides at 5 miles per hour. When the boys meet halfway between their houses, they will continue to the park together. How long will that take? How far will each boy have ridden?

In
hours, both boys will have ridden
miles.

Answer 1

Let's denote:

• \( t \): the time (in hours) that Preston has been biking when he meets Darren.
• \( d \): the total time (in hours) that Darren has been biking when they meet.

From the problem, we know:

1. Darren has already biked for a certain amount of time before Preston started. He has gone 2 miles at a speed of 3 miles/hour. The time \( d \) can be expressed as: \[ d = t + \frac{2}{3} \]

2. The distance Darren has traveled when they meet can be expressed as: \[ \text{Distance of Darren} = 3d = 3\left(t + \frac{2}{3}\right) \]

3. The distance Preston travels in time \( t \) at a speed of 5 miles/hour is: \[ \text{Distance of Preston} = 5t \]

Since they meet halfway between their houses, the distances they travel should be equal to half the total distance between their houses. Therefore, we need to set the two distances equal to each other.

So we can set up the equation:

\[ 3\left(t + \frac{2}{3}\right) = 5t \]

Now, let's solve this equation step-by-step.

1. Distributing 3: \[ 3t + 2 = 5t \]

2. Subtract \( 3t \) from both sides: \[ 2 = 2t \]

3. Divide by 2: \[ t = 1 \]

Now we can find the total time Darren has been biking:

\[ d = t + \frac{2}{3} = 1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3} \text{ hours} \]

Next, let's find the distances each boy has ridden when they meet.

For Darren: \[ \text{Distance of Darren} = 3d = 3 \times \frac{5}{3} = 5 \text{ miles} \]

For Preston: \[ \text{Distance of Preston} = 5t = 5 \times 1 = 5 \text{ miles} \]

Finally, the time they need to finish biking to the park together is simply their meeting time since they will continue together.

So, the time for both boys to meet is:

\[ t = 1 \text{ hour} \]

Now let's fill in the blanks:

In 1 hours, both boys will have ridden 5 miles.

Final Summary:

• Time until they meet: 1 hour
• Distance each boy rides by the time they meet: 5 miles.