SOLUTION: I am trying to help my daughter do her homework and it has been decades since I have done some of this. We are having a problem with a distance word problem. As a part of her tr

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Question 839479: I am trying to help my daughter do her homework and it has been decades since I have done some of this. We are having a problem with a distance word problem.
As a part of her training routine for basketball, Shaylee alternates between cycling and running. She cycles at a rate of 12mph and runs at a rate of 8mph. If she spends 5.5 hours and covers 50 miles, how much time did she spend on each exercise.
Many thanks in advance,
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +d+ = the distance in miles she cycled
+50+-+d+ = the distance in miles she ran
Let +t+ = the time in hours she spent cycled
+5.5+-+t+ = the time in hours she spent running
--------------------------------------------
Equation for cycling:
(1) +d+=+12t+
Equation for running:
(2) +50+-+d+=+8%2A%28+5.5+-+t+%29+
--------------------------
This is 2 equations and 2 unknowns, so it's solvable
(2) +50+-+d+=+44+-+8t+
Add +d+ to both sides
(2) +50+=+d+%2B+44+-+8t+
Subtract +8t+ from both sides
(2) +50+%2B+8t+=+d+%2B+44+
Subtract +44+ from both sides
(2) +d+=+8t+%2B+6+
Substitute (1) into (2)
(2) +12t+=+8t+%2B+6+
Subtract +8t+ from both sides
(2) +4t+=+6+
(2) +t+=+1.5+
and
+5.5+-+t+=+4+
and, since
(1) +d+=+12t+
(1) +d+=+12%2A1.5+
(1) +d+=+18+
and
+50+-+d+=+32+
-------------------
18 = the distance in miles she cycled
32 = the distance in miles she ran
1.5 = the time in hours she spent cycled
4 = the time in hours she spent running
check:
(2) +50+-+d+=+8%2A%28+5.5+-+t+%29+
(2) +50+-+18+=+8%2A%28+5.5+-+1.5+%29+
(2) +32+=+8%2A4+
(2) +32+=+32+
OK
(1) +d+=+12t+
(1) +18+=+12%2A1.5+
(1) +18+=+18+
OK