A company has a $150 budget to provide lunch for its 20 employees. The options are to provide either roast beef sandwiches, which cost $5 apiece, or tuna sandwiches, which also cost $5 apiece. The company also wants to use the entire budget. Suppose r represents the number of roast beef sandwiches it provides and t represents the number of tuna sandwiches. Which statement is correct?

Question

The company can provide lunch for all 20 employees and use the entire budget because there is a solution to the system of equations r minus t = 20 and 5 r + 5 t = 150.
The company can provide lunch for all 20 employees and use the entire budget because there is a solution to the system of equations r + t = 20 and 5 r + 5 t = 150.
The company cannot provide lunch for all 20 employees and use the entire budget because there is no solution to the system of equations r minus t = 20 and 5 r + 5 t = 150.
The company cannot provide lunch for all 20 employees and use the entire budget because there is no solution to the system of equations r + t = 20 and 5 r + 5 t = 150.

Answer 1

To analyze the problem, let's break down the two equations being presented and see which one correctly represents the situation.

1. Number of sandwiches equation: The company wants to provide lunch for all 20 employees, which means the total number of sandwiches (roast beef and tuna) should equal 20. This can be represented as: \[ r + t = 20 \]

2. Budget equation: The total cost of the sandwiches must equal $150, with each sandwich costing $5. Therefore, the cost equation is: \[ 5r + 5t = 150 \] We can simplify this by dividing every term by 5, resulting in: \[ r + t = 30 \]

Now, let's compare these two equations:

• From the first equation \( r + t = 20 \) represents the total number of sandwiches needed for the 20 employees.
• The second simplified equation \( r + t = 30 \) would imply that the company has enough budget to provide 30 sandwiches, which is not consistent with the requirement for only 20 employees.

To summarize, if we want to provide lunch for all 20 employees using the entire budget, we must have the equations:

1. \( r + t = 20 \) (total sandwiches for 20 employees)
2. \( r + t = 30 \) (the budget condition simplifies incorrectly)

Since \( r + t = 20 \) cannot occur simultaneously with \( r + t = 30 \), it leads to an inconsistency.

Thus, the correct statement is: The company cannot provide lunch for all 20 employees and use the entire budget because there is no solution to the system of equations \( r + t = 20 \) and \( 5r + 5t = 150 \).