A packaging company has been offered a contract to create gift boxes for perfume/cologne. The company needs to buy a special machine to make the boxes. The machine costs $3,000, and each box costs $2 for labor and materials. The perfume/cologne maker as agreed to buy each box for $5. (5 points)

Question

1.) The equation showing the packaging company's cost to make the boxes:
(hint: when entering your equation, DO NOT include spaces after each value or commas. Example: enter this: y=10x+7000, NOT y = 10 x + 7,000)

2.) The equation showing the perfume/cologne maker's cost to purchase the boxes:
(hint: when entering your equation, DO NOT include spaces after each value or commas. Example: enter this: y=10x+7000, NOT y = 10 x + 7,000)

3.) What is the solution to this system of equations?
(make sure to list your answer as an ordered pair; DO NOT add commas to numbers larger than 999)

4.) How many boxes does the perfume/cologne maker need to order for the packaging company to break even?
boxes

5.) How many boxes does the perfume/cologne maker need to order for the packaging company to make money?
boxes

Question 2
6.) Use this space to show your work for solving this system of equations. Make sure to show all the steps necessary to solve the system of equations. (5 points)

Answer 1

1.) The equation showing the packaging company's cost to make the boxes:
Cost to make each box = $2
Number of boxes = x
Total cost = 2x + 3000

2.) The equation showing the perfume/cologne maker's cost to purchase the boxes:
Cost to purchase each box = $5
Number of boxes = x
Total cost = 5x

3.) To find the solution to this system of equations, we need to set the costs equal to each other and solve for x:
2x + 3000 = 5x

Subtract 2x from both sides:
3000 = 3x

Divide both sides by 3:
x = 1000

So the solution to this system of equations is (1000, 1000).

4.) To break even, the total cost to make the boxes should be equal to the amount the perfume/cologne maker is willing to pay for each box. So we set the equations equal to each other:
2x + 3000 = 5x

Subtract 2x from both sides:
3000 = 3x

Divide both sides by 3:
x = 1000

So the perfume/cologne maker needs to order 1000 boxes for the packaging company to break even.

5.) The packaging company will make money if the total cost to make the boxes is less than the amount the perfume/cologne maker is willing to pay for each box. So we set the inequality:
2x + 3000 < 5x

Subtract 2x from both sides:
3000 < 3x

Divide both sides by 3:
x > 1000

So the perfume/cologne maker needs to order more than 1000 boxes for the packaging company to make money.

Question 2
6.) Solution:
We can solve the system of equations by subtracting 2x from both sides of the equation in step 3:
3000 = 3x - 2x

Combining like terms:
3000 = x

So the number of boxes needed is x = 3000. However, we need to check this solution to see if it is feasible. Plugging x = 3000 into either of the original equations:
2(3000) + 3000 = 5(3000)
6000 + 3000 = 15000

Therefore, the correct solution is x = 1000.