As a restaurant owner there are many decisions that you need to make on a daily basis, such as where to keep inventory levels. You wish to replenish your stock of dishes by purchasing 250 sets for your restaurant. You have two dish design from which to choose. One design costs $20 per set and the other $45 per set. If you only have $6,800 to spend, how many of each design should you order? •Solve the equations for the different dish designs to be ordered with the desired technique: graphing, substitution, elimination, matrix.•Explain how to check your solution for both equations.

Question

As a restaurant owner there are many decisions that you need to make on a daily basis, such as where to keep inventory levels. You wish to replenish your stock of dishes by purchasing 250 sets for your restaurant. You have two dish design from which to choose. One design costs $20 per set and the other $45 per set. If you only have $6,800 to spend, how many of each design should you order? •Solve the equations for the different dish designs to be ordered with the desired technique: graphing, substitution, elimination, matrix.•Explain how to check your solution for both equations.

Answer 1

1) 20x + 45y = 6800

2) 20(250 - y) + 45y = 6800 {multiply 20 through the parentheses}
3) 5000 - 20y + 45y = 6800 {combine the y terms 45y - 20y = 25y}
4) 5000 + 25y = 6800 {subtract 5000 from both sides}
5) 25y = 1800 {divide 25 from both sides}
6) y = 72

Now put y back into equation 1) x + y = 250

1) x + 72 = 250
2) x = 250 - 72
3) x = 178

Answer 2

To check, put both values back into the second equation.

20(178) + 45(72) = 6800

3560 + 3240 = 6800

Answer 3

To solve this problem, we can use the substitution technique. Let's assign variables to represent the number of sets of each design of dishes.

Let's say x represents the number of sets of the first design ($20 per set) and y represents the number of sets of the second design ($45 per set).

We have two equations:
1. x + y = 250 (the total number of sets needed)
2. 20x + 45y = 6,800 (the total cost available)

To solve the equations using the substitution technique, we need to isolate one variable in one of the equations and substitute it into the other equation.

Let's solve equation 1 for x:
x = 250 - y

Now we can substitute x in equation 2 with the above expression:
20(250 - y) + 45y = 6,800

Now simplify and solve the equation:
5,000 - 20y + 45y = 6,800
25y = 6,800 - 5,000
25y = 1,800
y = 1,800 / 25
y = 72

Now substitute the value of y into x = 250 - y:
x = 250 - 72
x = 178

So, you should order 178 sets of the first design and 72 sets of the second design.

To check your solution, substitute the values of x and y into both equations and see if they hold true.

Equation 1: x + y = 250
178 + 72 = 250
250 = 250

Equation 2: 20x + 45y = 6,800
20(178) + 45(72) = 6,800
3,560 + 3,240 = 6,800
6,800 = 6,800

Both equations hold true, so the solution is correct.