How do you set this up?
Solve the system using substitution:
A boat travels 18 miles downstream in 1.5 hrs. It then takes the boat 3 hrs. to travel upstream the same distance. Find the speed of the boat in still water and the speed of the current.
Thanks you!!!
Thank you for your help!
To solve this system using substitution, we first need to set up two equations based on the given information.
Let's say the speed of the boat in still water is represented by "b" and the speed of the current is represented by "c".
Equation 1: (Downstream) distance = speed × time
The boat travels 18 miles downstream in 1.5 hours, so the equation becomes:
18 = (b + c) × 1.5
Equation 2: (Upstream) distance = speed × time
The boat takes 3 hours to travel upstream the same distance, so the equation becomes:
18 = (b - c) × 3
Now we have a system of two equations:
18 = (b + c) × 1.5 --(Equation 1)
18 = (b - c) × 3 --(Equation 2)
To solve this system using substitution, we can rearrange Equation 1 to express "b" in terms of "c" and substitute it into Equation 2.
Step 1: Solve Equation 1 for "b".
Divide both sides of Equation 1 by 1.5:
18 ÷ 1.5 = b + c
12 = b + c
b = 12 - c
Step 2: Substitute the value of "b" from Step 1 into Equation 2.
18 = (12 - c - c) × 3
18 = (12 - 2c) × 3
18 = 36 - 6c
6c = 36 - 18
6c = 18
c = 18 ÷ 6
c = 3
Step 3: Substitute the value of "c" from Step 2 into Equation 1 to find "b".
b = 12 - c
b = 12 - 3
b = 9
Therefore, the speed of the boat in still water is 9 mph and the speed of the current is 3 mph.
To solve this system of equations using substitution, we first need to define our variables. Let's represent the speed of the boat in still water as "b" and the speed of the current as "c".
Now we can set up our equations using the given information:
Equation 1: Distance = Speed × Time
For the downstream journey:
Distance = 18 miles
Time = 1.5 hours
Speed = (boat's speed in still water) + (speed of the current)
So, the equation becomes: 18 = (b + c) × 1.5
Equation 2:
For the upstream journey:
Distance = 18 miles
Time = 3 hours
Speed = (boat's speed in still water) - (speed of the current)
So, the equation becomes: 18 = (b - c) × 3
Now, we have a system of two equations:
Equation 1: 18 = (b + c) × 1.5
Equation 2: 18 = (b - c) × 3
To solve this system using substitution, we can isolate one variable in one of the equations and substitute it into the other equation.
Let's solve Equation 1 for b:
Divide both sides of Equation 1 by 1.5:
(18 ÷ 1.5) = b + c
12 = b + c
b = 12 - c
Now, substitute the value of b into Equation 2:
18 = (b - c) × 3
18 = (12 - c - c) × 3
18 = (12 - 2c) × 3
18 = 36 - 6c
6c = 36 - 18
6c = 18
c = 18 ÷ 6
c = 3
Finally, substitute the value of c back into the equation b = 12 - c:
b = 12 - 3
b = 9
Therefore, the speed of the boat in still water is 9 mph and the speed of the current is 3 mph.
To summarize the steps:
1. Set up the equations using the given information.
2. Solve one equation for one variable.
3. Substitute the value of that variable into the second equation.
4. Solve for the remaining variable.
5. Substitute the value of the remaining variable back into the first equation to find its value.
6. Provide the solution.
Hope this helps!
if the boat's speed is b, and the current is c,
distance = time * speed
1.5(b+c) = 18
3(b-c) = 18
now go solve 'em.