We want to find the speed of the current in a river. We can create a system of linear equations and then solve it using the Elimination Method. In this situation, we will represent the speed of the powerboat as x, the speed of the current as y, and the overall speed that the boat travels as r. First, let's focus on an important fact regarding the speed of the powerboat.
Notice that the current and the powerboat are traveling in opposite directions, meaning that the current slows down the boat. Let's subtract the rate of speed of the current from the speed of the powerboat.
r= x- y
Next, let's focus on the return trip when the direction of the powerboat and the current are the same. Consider how this changes the picture of our boat and the current.
This time the current will speed up the boat, so we will add the speed of the current to the speed of the powerboat.
r= x+ y
Keeping this in mind, let's recall the distance formula.
d=rt
In this formula, r is the rate of speed, d is the distance traveled, and t is the time spent traveling. Now, let's focus on our system of equations. From the exercise, we know that the powerboat takes 30 minutes to travel 10 miles when it goes with the current. Let's substitute x+ y for r from our second equation and rewrite 30 minutes as 12 hour.
Formula
Substitution in Hours
Substitution in Minutes
d=r t
10=( x+ y) * 1/2
10=( x+ y) * 30
We also know that the return trip takes 50 minutes because the powerboat is traveling against the current. Let's use our first equation to substitute x- y for r and rewrite 50 minutes as 56 hour. The distance the boat travels is still 10 miles.
Formula
Substitution in Hours
Substitution in Minutes
d=r t
10=( x- y) * 5/6
10=( x- y) * 50
Now that we have two equations, we can combine them into a system of equations.
10=(x+y) 12 & (I) 10=(x-y) 56 & (II)
Let's rewrite the equations to eliminate the fractions. This will make it easier to solve the system.
Now we are ready to solve the system to find the speed of the current. Notice that there are two pairs of like terms that have the same or opposite coefficients.
20= x + y 12= x - y
If we subtract Equation (II) from Equation (I), we will get an equation in only one variable, y. Since we only want to know the speed of the current y, this is ideal. Let's do it!
