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In this example the exponents to be subtracted had the sign —, which in subtracting was changed to +.
When a question has been resolved generally, that is, by representing the known quantities by letters, we sometimes propose to determine what values the unknown quantities will
take, for particular suppositions made upon the known quantities.
The two following questions offer nearly all the circumstances that can ever occur in equations of the first degree. A C B
Two couriers set out at the same time from the points A and B, distant from each other a number m of miles, and travel towards each other until they meet. The courier who sets out from the point A, travels at the rate of a miles per hour; the other travels at the rate of b miles per hour. At what distance from the points A and B will they meet 2
Suppose C to be the point, and
Since the first courier travels a miles, at the rate of a miles
per hour, he will be * hours upon the road. The second cou0. - . . . .
rier will be # hours upon the road. But they travel equal
Since neither of the quantities in these values of w and y has the sign —, it is impossible for either value to become nega
tive. . Therefore whatever numbers may be put in place of a, b, and m, they will give an answer according to the conditions of the question. In fact, since they travel towards each other, whatever be the distance of the places, and at whatever rate they travel, they must necessarily meet. Suppose now that the two couriers setting out from the points A and B situated as before, both travel in the same direction towards D, at the same rates as before. At what distances o the points A and B will the place of their meeting, C, e :
The second equation expressing only the equality of the time will not be altered.
Here the values of a and y will not be positive unless a is greater than 5; that is, unless the courier, that sets out from A, travels faster than the other.
In this case the point C, where they come together, is distant from A twice the distance A B.
Suppose a smaller than b, for example a = 4 and b = 8.
Then •= ** =–m 4–8 8 m = — "o = — 2 m. y 4–8 Here the values of r and y are both negative; hence there is some absurdity in the enunciation of the question for these numbers. In fact, it is impossible that the courier setting out from A, and travelling slower than the other should overtake him. Let us put c and y negative in the two equations, that is, change their signs.
The second equation is not affected by changing the sign ; and it ought not to be so, since it expresses only the equality of the times.
The first equation becomes y—a = m, instead of a -y = m, which shows that the point where they are together is nearer to A than to B, by the distance from A to B. It must therefore be on the other side of A, as at E.
E A B C D
The enunciation of the question may be changed in two ways so as to answer the conditions of this equation. -
First, we may suppose, that the couriers, setting out from A and B, instead of going towards D, go in the opposite direction, the one from A at 4 miles per hour, and the other from B at 8 miles per hour ; at what distance from the points A and B is the point E, where they come together
Or we may suppose that two couriers setting out from the same place E, one travelling at the rate of 4 miles, and the other 8 per hour, have arrived at the same time at the points A and B, which are m miles asunder. What distance are the points A and B from E *
How is this result to be interpreted
Observe that in this case a and b being equal, the two couriers travel equally fast, it is therefore impossible that one should ever overtake the other, however far they may travel in either direction, and no change in the conditions can make it possible. Zero being divisor, then, is a sign of impossibility.
We may observe that when there is any difference, however small, between a and b, the values of 3 and y will be real, and the couriers will come together in one direction or the other and the smaller the difference, the greater will be the distance travelled before they come together; that is, the greater will be the values of r and y.