# System of 3 equations involving speed, distance and time - stumped!

• Oct 2nd 2012, 10:38 PM
mammothskip
System of 3 equations involving speed, distance and time - stumped!
I have only been able to set up the solution to this problem, but am completely stuck on how to proceed. All help is greatly appreciated.

The problem:
A bicyclist averages one speed uphill, one speed on level ground, and one speed downhill. She estimates the following mileage for her thrwe previous rides:

• 2 miles uphill, 15 miles level, 5 miles downhill - 1.5 hours

• 6 miles uphill, 9 miles level, 1 miles downhill - 1.4 hours

• 8 miles uphill, 3 miles level, 8 miles downhill - 1.6 hours

What were her average speeds uphill, on level ground and downhill?

My partial solution:
Using U, L and D for uphill, level and downhill speeds, respectively, here are the set of equations I have set up:

• 2/U + 15/L + 5/D = 1.5
• 6/U + 9/L + 1/D = 1.4
• 8/U + 3/L + 8/D = 1.6

Assuming this is right, now what?
• Oct 3rd 2012, 01:18 AM
chiro
Re: System of 3 equations involving speed, distance and time - stumped!
Hey mammothskip.

Let x = 1/U, y = 1/L and z = 1/D. Doing this you now have a linear system in the form of Ax = b where you have b and you have A.

Can you now find the inverse of A to get x = A^(-1)*b?
• Oct 3rd 2012, 08:52 AM
bjhopper
Re: System of 3 equations involving speed, distance and time - stumped!
Following chiro's simplification rewrite your three equations in x,y,z and solve by elimination.
Eliminate x from 1 and 2,then 1 and 3 to give two equations in y and z. Eliminate y or z from these two