<PRE><B>Question 167374</B>
here&#39;s what i think is the answer.
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let R = rate that Roberto travels at.
let J = rate that Juanita travels at.
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if they walk towards each other they will meet in 1 hour.
therefore the total distance they traveled is 7.7 miles.
since Rate * Time = Distance,
Roberto traveled for 1 hours at rate R, and Juanita traveled for 1 hour at rate J.
equation for that is:
1*R + 1*J = 7.7
which is the same as:
R + J = 7.7 (first original equation) ************************************
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if they walk in the same direction, then they will meet in 11 hours.
Roberto will have walked 7.7 miles more than Juanita.
therefore
11 hours * rate of Roberto = D + 7.7 
11 hours * rate of Juanita = D 
equation for that is:
11*R = D + 7.7  (second original equation) *************
11*J = D  (third original equation) ********************
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the total distance traveled by both would be 2*D + 7.7
so, .....
11*R + 11*J = 2D + 7.7
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we now have 2 equations that need to be solved simultaneously.
R + J = 7.7
11*R + 11*J = 2D + 7.7
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if R + J = 7.7, then 11*R + 11*J = 11 * 7.7
so, .....
11*R + 11*J = 11 * 7.7 = 84.7
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since 2D + 7.7 = 11*R + 11*J, and 84.7 = 11*R + 11*J, then 2D+7.7 = 84.7 because 2 things equal to the same thing are equal to each other.
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equation to solve is:
2D + 7.7 = 84.7
subtract 7.7 from both sides.
2D = 77
D = 38.5
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from above, we know that:
11*R = D + 7.7
11*J = D
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solvng for J, we get:
11*J = D = 38.5
divide both sides by 11.
D = 38.5/11 = 3.5 mph.
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solving for R, we get:
11*R = D + 7.7 = 38.5 + 7.7 = 46.2
divide both sides by 11.
R = 4.2 mph.
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answer appears to be:
R = 4.2 mph
J = 3.5 mph.
to prove, substitute in original equations:
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first original equation is:
R + J = 7.7
4.2 + 3.5 = 7.7 
first equation looks good.
second original equation is:
11*R = D + 7.7
11*4.2 = 38.5 + 7.7 = 46.2
46.2 = 46.2
second original equation checks out ok.
third original equation is:
11*J = D
11*3.5 = 38.5
38.5 = 38.5
third original equation checks out ok.
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you could also have solved these as 2 simultaneous equations:
R + J = 7.7
11R + 11J = 2D + 7.7
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you would get the same answer, because:
multiply first equation by 11 to get:
11R + 11J = 84.7
11R + 11J = 2D + 7.7
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subtract top equation from bottom equation to get:
0 = 2D + 7.7 - 84.7
add 84.7 to both sides of equation.
84.7 = 2D + 7.7
subtract 7.7 from both sides of equation.
77 = 2D
divide both sides of equation by 2.
38.5 = D
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same answer.
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