# The Drive to Sue's Party

## So what's the deal?

Bob and Tim are invited to Sue's house party and they both choose to drive there. Bob lives 50km away from her, but Tim lives 10km closer. If Bob drives at 60km/h and Tim drives at 40km/h, who will arrive first? At what point would one of them pass the other and who would that be? Describe their journeys.

## Given Variables:

x = time (hours)

y = distance from Sue's house (km)

## Equations in y-intercept form and standard form:

Bob's Drive:

y = -60x + 50 or 60x + y - 50 = 0

Tim's Drive:

y = -40x + 40 or 40x + y - 40 = 0

## Finding the Point of Intersection:

Because both lines pass through the P.O.I. we can put both equations' y values into one equation:

-60x + 50 = -40x + 40

50 - 40 = -40x + 60x

10 = 20x

10 / 20 = x

0.5 = x

(0.5, 0)

The P.O.I.'s x coordinate is 0.5. Now, to find the y value, we can sub in any of the equations to find y.

y = -60 (0.5) + 50

y = -30 + 50

y = 20

(0.5, 20)

Hence, the P.O.I. is at (0.5, 20). This means that Bob would pass Tim after half an hour, with 20km left between them and Sue's house.

## Conclusion and Significance

Bob and Tim are both 20km away from Sue's house at half an hour.

By solving this linear system, the following has become evident:

• From 0 minutes to 30 minutes, Tim is closer to Sue's house.
• At half an hour, Bob and Tim are both 20km away from Sue's house.
• Bob would pass Tim after half an hour, with 20km left between them and Sue's house.

• After half an hour, Bob is closer to Sue's house.

We can also see that:

• Bob arrives at 0.83333333333333333 hours, which is at about 50 minutes.
• Tim arrives at exactly one hour.