1) Infinite solutions:
Scenario: A surfboard shop sells two types of surfboards, wood and fiberglass. They sell a total of 10 surfboards each day. The wood surfboards cost $200 each and the fiberglass surfboards cost $300 each. Write a system of equations to represent this scenario.
Let x represent the number of wood surfboards sold and y represent the number of fiberglass surfboards sold.
The system of equations is:
x + y = 10
200x + 300y = total revenue
Since the first equation can be represented by y = 10 - x, we substitute this into the second equation:
200x + 300(10 - x) = total revenue
200x + 3000 - 300x = total revenue
-100x + 3000 = total revenue
x = 30 - 30y
In this scenario, there are infinite solutions because for any value of x between 0 and 10, there will always be a corresponding value of y that satisfies the system of equations.
2) No solution:
Scenario: A surfboard shop sells wood surfboards for $150 each and fiberglass surfboards for $250 each. The shop sells a total of 15 surfboards each day. Write a system of equations to represent this scenario.
Let x represent the number of wood surfboards sold and y represent the number of fiberglass surfboards sold.
The system of equations is:
x + y = 15
150x + 250y = total revenue
When we simplify the equations, we get:
150x + 250(15 - x) = total revenue
150x + 3750 - 250x = total revenue
-100x + 3750 = total revenue
x = 37.5 - 37.5y
There is no solution in this scenario because the number of wood surfboards sold must be a whole number, but x = 37.5 which is not possible in this context.
3) One solution:
Scenario: A surfboard shop sells two types of surfboards, beginner and advanced. The shop sells a total of 20 surfboards each day. The beginner surfboards cost $100 each and the advanced surfboards cost $150 each. Write a system of equations to represent this scenario.
Let x represent the number of beginner surfboards sold and y represent the number of advanced surfboards sold.
The system of equations is:
x + y = 20
100x + 150y = total revenue
When we simplify the equations, we get:
100x + 150(20 - x) = total revenue
100x + 3000 - 150x = total revenue
-50x + 3000 = total revenue
x = 60 - 60y
In this scenario, there is one unique solution because there is only one combination of x and y values that satisfy the system of equations. For example, if x = 10, then y = 10 as well, resulting in a total revenue of $2500. make this like a 8th grader wrote it
Scenario: A surfboard shop sells two types of surfboards, beginner and advanced. The shop sells a total of 20 surfboards each day. The beginner surfboards cost $100 each and the advanced surfboards cost $150 each.
Let's say x represents the number of beginner surfboards sold and y represents the number of advanced surfboards sold. Here are the equations we came up with:
1) x + y = 20
2) 100x + 150y = total revenue
We simplified the equations like this:
1) x + y = 20
2) 100x + 150(20 - x) = total revenue
3) 100x + 3000 - 150x = total revenue
4) -50x + 3000 = total revenue
5) x = 60 - 60y
Our calculations led us to find a unique solution. For example, if the shop sells 10 beginner and 10 advanced surfboards, the total revenue would be $2500.