How Do You Solve Simultaneous Equations for Time and Distance in a Car Journey?

[AN630078]

Homework Statement

Hello, I have been revising mechanics problems when I came across the question below. I do not know whether my approach would here be suitable and I typically struggle with these sorts of questions, not necessarily solving the simultaneous equations but formulating them to begin with. I have answered it fully but I would appreciate if anyone could comment upon whether my workings and methodology are applicable here. Also, the units given in the original question are miles, hours and mph, should I convert these to km or m and seconds or would this be unnecessary?

A car journey of 200 miles lasts 4 hours. It is partially spent on the motorway travelling at 70mph and the remainder on country roads travelling at 40mph.

Write this information as a pair of simultaneous equations and find the distances travelled on each road.

Relevant Equations

speed=distance/time

Equation 1:
Where t1=time spent on motorway
Where t2=time spent on country roads

t1+t2=4

Equation 2:
Using distance = speed * time
200 = 70*t1+40*t2

Rearrange equation 1 in terms of t1;
t1=4-t2
Substitute the rearranged form of equation 1 into equation 2:
200=70(4-t2)+40t2
200=280-70t2+40t2
200=280-30t2
Rearrange to find t2:
30t2=280-200
30t2=80
t2=80/30=8/3 hours (This is equation 3)
(Would units of hours be appropriate here?)

Substitute equation 3 into the original form of equation 1 to find t1:
t1+8/3=4
t1=4-8/3
t1=4/3 hours

Since I have now found t1 and t2 I can substitute this information to find the distance traveled on each of the roads;
distance=speed*time
distance on the motorway=70*4/3=280/3~93.3 miles to 3.s.f
distance on the country roads=70*8/3=320/3~106.7 miles to 3.s.f

 

[Mark44]

A car journey of 200 miles lasts 4 hours. It is partially spent on the motorway traveling at 70mph and the remainder on country roads traveling at 40mph.

Write this information as a pair of simultaneous equations and find the distances traveled on each road.

Equation 1:
Where t1=time spent on motorway
Where t2=time spent on country roads

t1+t2=4
Equation 2:
Using distance = speed * time
200 = 70*t1+40*t2
Rearrange equation 1 in terms of t1;
t1=4-t2
Substitute the rearranged form of equation 1 into equation 2:
200=70(4-t2)+40t2
200=280-70t2+40t2
200=280-30t2
Rearrange to find t2:
30t2=280-200
30t2=80
t2=80/30=8/3 hours (This is equation 3)
(Would units of hours be appropriate here?)

Substitute equation 3 into the original form of equation 1 to find t1:
t1+8/3=4
t1=4-8/3
t1=4/3 hours

Since I have now found t1 and t2 I can substitute this information to find the distance traveled on each of the roads;
distance=speed*time
distance on the motorway=70*4/3=280/3~93.3 miles to 3.s.f
distance on the country roads=70*8/3=320/3~106.7 miles to 3.s.f

Looks good.

Another way to do the problem is by the use of a single variable.
Let $t_{70}$ = the time spent traveling at 70 mph
Then $4 - 4_{70}$ = the time spent traveling at 40 mph
This is obtained from the fact that the two times add up to 4 hours.

The equation is $70t_{70} + 4(4 - t_{70}) = 200$
Solving this equation produces the same values you got.

 

[AN630078]

Looks good.

Another way to do the problem is by the use of a single variable.
Let $t_{70}$ = the time spent traveling at 70 mph
Then $4 - 4_{70}$ = the time spent traveling at 40 mph
This is obtained from the fact that the two times add up to 4 hours.

The equation is $70t_{70} + 4(4 - t_{70}) = 200$
Solving this equation produces the same values you got.

Thank you very much for your reply and for your advice, I greatly appreciate it. I will certainly endevour to try your suggestion of using a single variable in future problems.

 

[Ibix]

Given that you are asked for distances, you could note that time is distance over speed and write equations in terms of the distances $d_{40}$ and $d_{70}$:$$\begin{eqnarray*}
200&=&d_{40}+d_{70}\\
4&=&\frac{d_{40}}{40}+\frac{d_{70}}{70}
\end{eqnarray*}$$This just saves a bit of mucking around at the end.

 

[Mark44]

I will certainly endevour to try your suggestion of using a single variable in future problems.

The problem asked you to write a pair of equations, which you did. I was just showing you an alternate way of doing things.

Also, in one of your solutions, you asked if you should include units. Yes, you should.

 

[AN630078]

The problem asked you to write a pair of equations, which you did. I was just showing you an alternate way of doing things.

Also, in one of your solutions, you asked if you should include units. Yes, you should.

Ok splendid, thank you again for your help