on a canoe trip, Rita paddled upstream (against the current) at an average speed of 2 mi/hr relative to the riverbank. on the return trip downstream (with the current), her average speed was 3 mi/hr. find rita's paddling speed in still water and the speed of the river's current.
dealing w/- substitution and elimination
let w be the water speed
let b be the boat speed in still water.
b-w=2
b+w=3
solve. I would add the two equations to eliminate w.
b-w=2
b+w=3 (transpose)
b=3-w (substitute)
3-w-w=2 (combine like terms)
-2w=-1 (cancel -2)
w= 1/2
To solve the problem using the method of substitution and elimination, let's consider the following variables:
Let the speed of the canoe in still water be represented by "c" (in mi/hr).
Let the speed of the river's current be represented by "r" (in mi/hr).
Now, let's use the given information to form two equations that describe the relationship between Rita's speed and the speed of the current.
1. When paddling upstream (against the current), Rita's speed is reduced by the speed of the current:
Rita's speed upstream = c - r = 2 mi/hr
2. When paddling downstream (with the current), Rita's speed is increased by the speed of the current:
Rita's speed downstream = c + r = 3 mi/hr
Now, we have a system of two equations. We can solve for "c" and "r" using either the substitution or elimination method.
Method 1: Substitution
Solve Equation 1 for "c":
c = 2 + r
Substitute this value of "c" into Equation 2:
(2 + r) + r = 3 mi/hr
2r + 2 = 3
2r = 3 - 2
2r = 1
r = 1/2
Substitute the value of "r" back into Equation 1 to find the value of "c":
c = 2 + (1/2) = 2.5 mi/hr
Therefore, Rita's paddling speed in still water is 2.5 mi/hr, and the speed of the river's current is 1/2 mi/hr.
Method 2: Elimination
Multiply Equation 1 by -1:
-(c - r) = -2 mi/hr
-r + c = -2
Now, add this equation to Equation 2:
(c + r) + (-r + c) = 3 - 2
2c = 1
c = 1/2
Substitute the value of "c" back into Equation 1 to find the value of "r":
1/2 - r = 2 mi/hr
-r = 2 - 1/2
-r = 3/2
r = -3/2
However, since we are dealing with speed, we cannot have a negative value. Hence, we discard this solution.
Therefore, Rita's paddling speed in still water is 1/2 mi/hr, and the speed of the river's current is 1/2 mi/hr.
To find Rita's paddling speed in still water and the speed of the river's current, we will use a system of equations and solve it using either substitution or elimination method.
Let's represent the paddling speed in still water as "x" and the speed of the river's current as "y".
According to the given information:
- On the upstream trip, Rita paddled at an average speed of 2 mi/hr relative to the riverbank. This means her effective speed was reduced by the speed of the current, so her actual speed was x - y.
- On the downstream trip, Rita paddled at an average speed of 3 mi/hr. In this case, her effective speed was increased by the speed of the current, so her actual speed was x + y.
Therefore, we have the following two equations:
Equation 1: x - y = 2
Equation 2: x + y = 3
Now, we can solve this system of equations using either substitution or elimination method.
Substitution Method:
In this method, we solve one equation for one variable and substitute it into the other equation.
From Equation 2, we can express x in terms of y:
x = 3 - y
Substituting this value of x into Equation 1:
(3 - y) - y = 2
3 - 2y = 2
-2y = -1
y = 1/2
Now, we substitute the value of y back into Equation 2 to find the value of x:
x + (1/2) = 3
x = 3 - (1/2)
x = 5/2
Therefore, Rita's paddling speed in still water is 5/2 mi/hr, and the speed of the river's current is 1/2 mi/hr.
Using the Elimination method:
In this method, we eliminate one variable by adding or subtracting the equations.
To eliminate the variable "y", we can add Equation 1 and Equation 2:
(x - y) + (x + y) = 2 + 3
2x = 5
x = 5/2
Now, substituting the value of x back into Equation 1:
(5/2) - y = 2
- y = 2 - (5/2)
- y = 1/2
y = -1/2
Again, we find that Rita's paddling speed in still water is 5/2 mi/hr, and the speed of the river's current is 1/2 mi/hr.
Therefore, both the substitution and elimination methods give the same solution.