solving a rate problem using system of linear equations

[bhakimjavadi399]

Two cyclists start at the same point and travel in opposite directions. One cyclist travels
5 km/h slower than the other. If the two cyclists are 212 km apart after 4 hours, what is the rate of each cyclist?

 

[MarkFL]

For the speedier cyclist:

[MATH]d_1=4v[/MATH]
And for the slower cyclist:

[MATH]d_2=4(v-5)[/MATH]
What is the sum of their distances traveled?

 

[bhakimjavadi399]

Hi,
I need to find the rate , not distance

 

[MarkFL]

Distance, speed and time are interrelated, and it is through this relationship that we can find one, if we know the others. You are actually given the sum of the distances traveled by the cyclists:

[MATH]d_1+d_2=212[/MATH]
And now we have 3 equations in 3 unknowns. What do you get if you add the first two equations I gave?

 

[bhakimjavadi399]

212/4 = 53

 

[bhakimjavadi399]

i need more help plz

 

[MarkFL]

212/4 = 53

That would in fact be the sum of their speeds, but that's not what I meant. When we add the two equations I initially gave, we get:

[MATH]d_1+d_2=4v+4(v-5)[/MATH]
Use the third equation I gave to replace the sum \(d_1+d_2\), and then you'll have an equation in \(v\) alone which you can solve, and whose value will let you determine the speeds of both cyclists, since the faster cyclist's speed will be \(v\) and the slower cyclist's speed will be \(v-5\).

 

[bhakimjavadi399]

v=29?

 

[bhakimjavadi399]

speedier cycle: 116
slower : 111
? is that correct?

 

[Harry_the_cat]

I usually set up these speed/distance/time questions in a table. Let v be the speed of the first cyclist and x be the distance travelled by the first cyclist.
Then you can form 2 equations with 2 unknowns, and solve:
\(\displaystyle v=\frac{x}{4}\)
and
\(\displaystyle v-5 = \frac{212-x}{4}\)

	Speed	Distance	Time
Cyclist 1	v	x	4
Cyclist 2	v-5	212-x	4

 

[MarkFL]

v=29?

Yes.