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
Real-world scenario:
You are a manager of a coffee shop and you are trying to decide how many regular and decaf coffee bags to order for the upcoming week. The regular coffee costs $5 each and the decaf coffee costs $4 each. You need a total of 40 bags of coffee and you have a budget of $180. Write a system of equations to represent this scenario and solve for the number of regular and decaf coffee bags to order.
Let x represent the number of regular coffee bags and y represent the number of decaf coffee bags.
Equation 1: x + y = 40 (total number of bags)
Equation 2: 5x + 4y = 180 (total cost)
Solving the system of equations:
Equation 1: x + y = 40
Equation 2: 5x + 4y = 180
Multiply Equation 1 by 4:
4x + 4y = 160
Subtract the modified Equation 1 from Equation 2:
5x + 4y - 4x - 4y = 180 - 160
x = 20
Substitute x = 20 into Equation 1:
20 + y = 40
y = 20
Therefore, the manager should order 20 bags of regular coffee and 20 bags of decaf coffee to stay within budget and fulfill the total number of bags needed.
make it so it is a diffrent senaro
Real-world scenario:
You are a music store manager and you are trying to decide how many CDs and vinyl records to order for a new release from a popular artist. The CDs cost $10 each and the vinyl records cost $15 each. You need a total of 50 items to stock and you have a budget of $600. Write a system of equations to represent this scenario and solve for the number of CDs and vinyl records to order.
Let x represent the number of CDs and y represent the number of vinyl records.
Equation 1: x + y = 50 (total number of items)
Equation 2: 10x + 15y = 600 (total cost)
Solving the system of equations:
Equation 1: x + y = 50
Equation 2: 10x + 15y = 600
Multiply Equation 1 by 10:
10x + 10y = 500
Subtract the modified Equation 1 from Equation 2:
10x + 15y - 10x - 10y = 600 - 500
5y = 100
y = 20
Substitute y = 20 into Equation 1:
x + 20 = 50
x = 30
Therefore, the manager should order 30 CDs and 20 vinyl records to have a total of 50 items in stock and stay within the budget.