Question

Michael has $15 and wants to buy a combination of school lunches to feed at least three classmates. A sandwich costs $2, and hot lunch costs $3. This system of inequalities models the scenario:

2x + 3y ≤ 15
x + y ≥ 3

Part A: Describe the graph of the system of inequalities, including shading and the types of lines graphed. Provide a description of the solution set.

Part B: Is the point (5, 1) included in the solution area for the system? Justify your answer mathematically.

Part C: Choose a point in the solution set and interpret what it means in terms of the real-world context.

Answer 1

The graph consists of two solid lines representing the constraints where Michael can spend $15 on sandwiches and hot lunches to feed at least three classmates. The solution set is the overlapping shaded area. The point (5, 1) is within the solution set, and an example point (3, 2) means Michael buys 3 sandwiches and 2 hot lunches.

Step-by-step explanation:

Part A: To graph the system of inequalities 2x + 3y ≤ 15 and x + y ≥ 3, you use a two-dimensional coordinate system. For the first inequality, you would draw a straight line corresponding to the equation 2x + 3y = 15, which represents the boundary where Michael spends exactly $15. This line would be solid because the inequality includes values that are equal to 15 as well. The area below and including the line is shaded to represent all the combinations of x (sandwiches) and y (hot lunches) that cost at most $15. For the second inequality, a straight line will be drawn for the equation x + y = 3, representing the fewest number of lunches Michael can buy for his three classmates. As this line represents at least three lunches, it will also be solid, and the area above and including the line will be shaded. The solution set consists of the area where both shadings overlap.

Part B: To determine if the point (5, 1) is included in the solution area, substitute x with 5 and y with 1 into both inequalities. 2(5) + 3(1) ≤ 15 simplifies to 10 + 3 ≤ 15, which is true. The second inequality, 5 + 1 ≥ 3, simplifies to 6 ≥ 3, which is also true. Therefore, the point (5, 1) is in the solution set.

Part C: Choosing the point (3, 2) from the solution set means Michael buys 3 sandwiches and 2 hot lunches. This combination respects both constraints because 2(3) + 3(2) = 12 ≤ 15 and 3 + 2 = 5 ≥ 3, so it fits his budget and feeds at least three classmates.

Answer 2

Step-by-step explanation:

A-the first equation which is 2x + 3y ≤ 15 is shaded downwards while the other equation x + y ≥ 3 is shaded upwards

B-(5,1) is included in the solution area for the 2x+3y<15 equation because when x and y are substituted ,it is still less than 15

so 5+1 is greater than 3

C-(2,3) MICHEAL can buy 2 sandwiches and 3 hot lunches which is Whitin his 15 dollar budget and can feed at least 3 of his classmates ]

i hope this is correct I'm not sure 100% especially part c if it incorrect put comment with correction