SOLUTION: A tourist sailed against the current on a river for 6 km, and then he sailed in a lake for 15 km. In the lake he sailed for 1 hour longer than he sailed in the river. Knowing that

Question 969357: A tourist sailed against the current on a river for 6 km, and then he sailed in a lake for 15 km. In the lake he sailed for 1 hour longer than he sailed in the river. Knowing that the current of the river is 2 km/hour, find the speed of the boat while it is traveling in the lake
6/(x-2) = 15/x +1
?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
it looks like you had the right idea to start with but maybe didn't know how to finish it.

the general formula is r * t = d

r is the rate of speed.
t is the time.
d is the distance.

the formula in the river becomes:

(r-2) * t = 6

the formula in the lake becomes:

r * (t + 1) = 15

you have 2 equations tht need to be solved simultaneously because the same value of r and the same value of t applies to both equations.

those equation are:

(r-2) * t = 6
r * (t + 1) = 15

there are various ways to solve two equations simultaneously.
we'll do it the way that is shown below:
just keep in mind that this isn't the only way that simultensous equations can be solved.

there substitution and elimination and graphing.
those are the big three.
this way is by elimination.
you can also solve both equations for the same variable and then set those equations equal to each other.
the whole idea is to eliminate one of the variables so the other variable can be solved for.
once that variable is solved for, you can use that value to solve for the other variable.

simplify these equations to get:

rt - 2t = 6
rt + r = 15

subtract the first equation from the second equation to get:

r + 2t = 9

solve for r to get r = 9 - 2t.

go back to the first original equation and replace r with 9 - 2t to get (r-2)*t = 6 to become (9 - 2t - 2) * t = 6

simplify that to get 9t - 2t^2 - 2t = 6

simplify further to get 7t - 2t^2 = 6

add 2t^2 to both sides of the equation and subtract 7t from both sides of the equation to get 0 = 2t^2 - 7t + 6 which is the same as 2t^2 - 7t + 6 = 0

factor that quadratic equation to get t - 2 or t = 3/2.

solve for r in the first original equation and you will get r = 5 or r = 6.

solve for r in the second original equation and you will get r = 5 or r = 6 as well.

your solutions are:

when t = 2, r = 5
when t = 3/2, r = 6

both solutions look good.

use them in both original equations and those equations should be true.

when t = 2 and r = 5, both original equations become:

(r-2)*t = 6 becomes (5-2)*2 = 6 which becomes 3*2 = 6 which becomes 6 = 6 which is true.

r*(t+1) = 15 becomes 5*(2+1) = 15 which becomes 15 = 15 which is true.

when t = 1.5 and r = 6, both original equations becomes:

(r-2)*t = 6 becomes (6-2)*1.5 = 6 which becomes 4*1.5 = 6 which becomes 6 = 6 which is true.

r*(t+1) = 15 becomes 6*(1.5+1) = 15 which becomes 6*2.5 = 15 which becomes 15 = 15 which is true.

both solutions are good.

the problem statement was:

find the speed of the boat while it is traveling in the lake.

the answer is that the speed of the boat when it is traveling in the lake is either 5 or 6 kmph.

it could be either one, unless i did something drastically wrong.

i also solved it graphically and got the same result.

there are two possible solutions as i described above.

to graph these equations, let y = t and let x = r and solve for y in both eqautions.

you will get:

y = 6/(x-2) and y = 15/x - 1

these are pretty close to what you had, except you had 15/x + 1 which was incorrect.

the graph would look something like this:

it's hard to see on this graph, but there are 2 intersection points at x = 6 and x = 6.

when x = 5, the value of y is 2.

when x = 6, the value of y is 1.5

the horizontal lines are the values of y = 2 and y = 1.5.

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