Now try writing and solving your own systems of linear equations: one with infinite solutions, one with no solution, and one with one solution. Write a real-world scenario that goes with at least one system of equations. Show your work

Question

make the scenario about a surfboard shop

Answer 1

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.