Question 536983
This looks hinky, but is actually pretty straightforward.  We need to recall the basic formula linking distance (d), time (t), and rate (r):
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d = r * t
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It is useful to think of this as two trips, one that is and one that could be.  In both trips, d = 504 miles.  We don't know r or t in either trip, but we are given the relationship between the two trips, so we can calculate r and t:
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in the case of multiple unknowns, it is best solved with a system of equations with as many equations as unknowns.  In this case two unknowns: r and t, so we need to derive two equations.  The easiest method is often substitution, in which in one of the equations, the equation needs to be written (or re-written) in terms of one of the unknowns.
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We are given: 
d = 504.  We know this is for both equal distance trips.
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Lets rewrite d = r * t in terms of r:
r = d / t
r = 504/t
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The 2nd equation relates the 2nd trip to the 1st: 7 mph faster shaves 1 hr off the time:
r + 7 and t - 1
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2nd equation: 
d = (r + 7) (t - 1)
504 = (r + 7)(t - 1)
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Substitute r = 504/t into this 2nd equation:
504 = (504/t + 7)(t - 1)
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Simplify the terms:
504 = 504t/t - 504/t + 7t - 7
505 = 504 - 504/t + 7t - 7
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Subtract 504 from both sides:
0 = 7t - 504/t  - 7
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Multiply both sides by t (so we can get rid of the 504/t term):
0 = 7t^2 - 7t - 504
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If we divide everything by 7 to simplify further, we end up with a standard form quadratic equation:
0 = t^2 - t - 72
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Factor the quadratic equation:
0 = (t + 8)(t - 9)
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 t = -8 or 9 are the possible solutions to the quadratic equation.  t = -8 is a nonsensical answer, so we can reject it.  t = 9 hrs is how long the original trip takes.  To figure out r, we just go back to the original equation with d = 504 and t = 9:
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d = r * t
504 = 9r
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divide both sides by 9:
504/9 = 9r/9
56 = r
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The original trip of 504 miles takes 9 hours at 56 mph.
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The notional trip means going 7 mph faster and shaving 1 hr off the time:
2nd trip: d = 504, r = 56 + 7, t = 9 - 1
r = 63 mph, t = 8 hrs
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Always check your work:
56 * 9 = 504 checks
63 * 8 = 504 checks
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The average speed for the notional trip is r = 63 mph
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Cheers,
Lee