Their speeds can be equal or different. Examine the situation depending on this fact.
B
b
Notice that the graphs are parallel.
C
c
Solve the system by writing an equation which J(t) is equal to M(t).
D
d
Solve the system by writing an equation which J(t) is equal to M(t).
E
e
Notice that Jorge did not start the trip.
A
a
No unique solution
B
b
No unique solution
C
c
Unique solution
D
d
Unique solution
E
e
No unique solution
Practice makes perfect
a
Two friends, Jorge and Mark, are taking a trip. They separately record the distance they each have traveled from home every hour on the second day. We will write a system of equations for each situation and check whether the system has a unique solution. The first situation is the following.
Jorge and Mark each leave at the same time.
Let J(t) be the distance traveled by Jorge and M(t) be the distance traveled by Mark. If we assume j is the speed of Jorge and m is the speed of Mark , we can write the following system depending on time t.
J(t)= jt & (I) M(t)= mt & (II)Here we have two different cases. Do not worry! We will examine them one by one.
Case I: j= m
If their speeds are the same, each t value will satisfy the system. Therefore, there are infinitely many solutions rather than an unique solution. We can show this by drawing a graph as the following.
As you can see their graph overlapping, which means there are infinitely many solutions.
Case II: j≠m
If their speeds are the same, there will not be any time t that satisfies the system. Therefore, there is no solution for the system. Let's show this visually.
As you can see, the only intersection point of their graphs is the leaving time t=0. Since they do not travel at t=0, it is not a solution. Their graph does not intersect after t=0. Therefore, there is no solution.
b
Let's continue with the second situation.
Jorge travels 25 miles on the first day and
drives 65 miles per hour on the second day.
Mark travels 100 miles on the first day and
also drives 65 miles per hour on the second day.
If we model the situation, their first day trips will be the y-intercept and their speeds will be the slope. Then, let's write the system.
J(t)= 65t+ 25 & (I) M(t)= 65t+ 100 & (II)
The slopes are the same for both equations, which means their graphs are parallel. Therefore, there is no solution.
c
The third situation is given below.
Jorge travels 25 miles on the first day and
drives 65 miles per hour on the second day.
Mark travels 100 miles on the first day and
also drives 45 miles per hour on the second day.
Let's model the situation as we did in Part B.
J(t)= 65t+ 25 & (I) M(t)= 45t+ 100 & (II)
Since the slopes and the y-intercepts are different, we assume that their graphs will intersect at one point. Therefore, There will be an unique solution for the system. In order to verify our assumption, let's solve the system by writing an equation which J(t) is equal to M(t).
J(t)&=M(t)
65t+ 25&= 45t+ 100
Let's solve it for t.
Jorge traveled 45 miles on the first day and
drives 55 miles per hour on the second day.
Mark arrived at the campsite on the first day
after traveling 300 miles.Let's write the system.
J(t)= 55t+ 45 & (I) M(t)= 300 & (II)
Again, since the slopes and the y-intercepts are different, we can say that their graphs intersects at one point. Therefore, there is an unique solution for the system. Let's find the solution to check whether it is positive.
Thus, we can say that the system has a positive unique solution.
e
Finally, we will decide the solution of the last situation.
Jorge gets sick before the trip
and doesn’t get in touch with Mark.
Mark travels 40 miles on the first day and
drives 67 miles per hour on the second day.
We can model this function as the following.
J(t)=0 & (I) M(t)= 67t+ 40 & (II)
As you can see, Jorge did not start the trip. Therefore, their graph never intersect, which means there is no solution.
