Answer:
1) S = 7, L = 19
2) Speed of the boat in still water = 9 mph, speed of the river water = 3 mph
Step-by-step explanation:
1) We need to represent this numerically. Let's let S = smaller number, L = larger number. Keep in mind the definitions of these words:
L = 5 + 2S
L - S = 12
Now let's use substitution to plug the first equation into the second equation:
L - S = 12
(5 + 2S) - S = 12
Solve for S:
5 + 2S - S = 12
5 + S = 12
S = 7
Plug S back into the equation to find L:
L - 7 = 12
L = 12 + 7
L = 19
2) Let's model this with some equations.
It takes the boat 2h to travel 24 miles downstream. That means that the boat is going with the current, and their speeds are being added together.
The speed of the boat + the speed of the water = 24 miles / 2 h = 12 mph
It takes the boat 3h to travel 18 miles upstream. That means that the boat is going against the current, so it takes longer and goes less far. Their speeds are going in opposite directions, so we need to subtract.
The speed of the boat - the speed of the water = 18 miles / 3 h = 6 mph
Let's represent the speed of the boat as B and the speed of the water as W and solve our system.
B + W = 12
B - W = 6
Subtract the two equations using elimination:
B + W = 12
- B - W = 6
= 0 + W - - W = 12 - 6
2W = 6
W = 3 mph
Plug this back in to get B, the speed of the boat:
B + W = 12
B + 3 = 12
B = 12 - 3 = 9 mph
Question 1: larger = 19, smaller = 7
The larger of two numbers is 5 more than twice the smaller. If the smaller is subtracted from the larger, the result is 12. Find the numbers.
To solve this question, you can represent the two equations using variables and algebra to solve it:
[tex]y = 2x + 5\\y - x = 12[/tex]
To solve this, you can substitute, or "plug in", the value for y stated in the first equation to the second one to solve for x:
[tex](2x+5)-x=12\\x = 7[/tex]
Knowing that x = 7, you can plug in that value of x into one of the equations to find y:
[tex]y = 2(7) + 5\\y = 19[/tex]
Question 2: the speed of the boat is 9 miles per hour, and the speed of the water is 3 miles per hour.
It takes a boat 2 hours to travel 24 miles downstream and 3 hours to travel 18 miles upstream. What is the speed of the boat in still water and the speed of the water of the river?
To find this, you can represent the speeds of the boat and the river using equations, where b is the speed of the boat and r is the speed of the river current:
[tex]b + r = 12\\b - r = 6[/tex]
(you can find the 12 and 6 values by dividing the miles by how many hours it takes, so 24/2 and 18/3)
You can use the same method for the last problem by solving for one variable:
[tex]b+r=12\\b = r+6[/tex]
Then, substitute it:
[tex](r+6)+r=12\\r=3[/tex]
Then, plug it into one of the equations to find that b = 9.
