Question 243381
Always be careful to define your variables and equations with the fewest possible unknowns.  That usually means defining variables in terms of one other wherever possible.
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In this case, we could say the speed of the plane is 'x' and that the speed of the train is 'x - 100', which means it is 100 mph slower.  OR we could say the train's speed is 'x' and the plane's speed = 'x + 100'.
Pick either one you prefer.
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Distance and rate problems are common in algebra.  Remember the fundamental equation is:
distance = rate * time
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The plane's characteristics are:
d = 500 km
r = x
t = same time as the train
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The train's characteristics are:
d = 300 km
r = x - 100
t = same time as plane
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Since the one thing that is equal is time, we can format each equation to isolate t.
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For the plane:
t = d/r
t = 500/x
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For the train:
t = d/r
t = 300/(x-100)
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Now we can rely on the principle that if a=b and b=c, then a=c.  Or "equals are equal."
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500/x = 300/(x-100)
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Now we can cross-multiply
500 * (x-100) = 300 * x
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Multiplying through on the left we have
500x - 50,000 = 300x
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Subtracting 300x from both sides
200x -50,000 = 0
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Adding 50,000 to both sides
200x = 50,000
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Dividing both sides by 200
x = 250
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Looking back at the setup, the train's speed is 250 - 100 = 150.
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Now we need to check our work.
How long does it take for the plane to travel 500 km?  2 hrs @ 250 km/hr
How long does it take the train to travel 300 km? 2 hrs @ 150 km/hr
So, we're confident in our answer:
The plane's speed is 250 km/hr, and the train's speed is 150 km/hr.
Done.