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# I have a linear question

Rob and Juanita are 7.7 miles apart.   If they walk in the same direction Rob will reach Juanita in 11 hours.  If they walk towards each other they meet in 1 hour.  What speed are they walking.

------------------------------------------------------------------------->,   rob's distance

----------------------------------->,   juan's distance

<----------------J---------------->

--------------------><---------------

<---7.7-x---------><-------x------>

<---------------7.7 mi------------->

rob's distance          juan's distance

We know both distances, and we know both times, so r = d/t

For rob:  (7.7 + J)/11 = (7.7-x)/1                       for juan:  J/11 = x/1

Solve by substitution,  x = 3.5  This is the distance juan walks toward rob.  Juan' speed = 3.5 mph

7.7 - 3.5 = 4.2   Robs speed = 4.2 mph

If two objects are moving toward to each other (or in the opposite directions), the velocity of convergence (or breakaway) is the sum of their individual velocities. d = (v1 + v2)*t (d - distance in miles, t - time in hours) ....... (1)
If the objects are moving in the same direction, then the velocity of convergence (or breakaway) is equal to the difference between their individual velocities. d = (v1 - v2)*t ........(2)
~~~~~~~~~~~~~~~~
Let's assume that Rob's velocity is vR and Juanita's velocity is vJ .
(1) If they walk towards each other they meet in 1 hour and distance they walked is 7.7 mi ------> (vR + vJ) * 1 = 7.7 ------> vR + vj = 7.7 --------> vR = 7.7 - vJ........(3)
(2) If they walk in the same direction Rob will reach Juanita in 11 hours  ------>
Rob's distance is          (11 * vR
—                -------> 11(    vR      - vJ) = 7.7
Juanita's distance is      (11 * vJ)                         ↓
7.7                              ↓
11((7.7 - vJ) - vJ) = 7.7
11                         11
7.7     - 2vJ    = 0.7 ------>
-2vJ = -7 -------> vJ = 3.5 mph ------> (3) ------> vR = 7.7 - 3.5 = 4.2 mph