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The complete question is:
In order to prepare for your summer bash, you go to the supermarket to buy hamburgers and chicken. Hamburgers cost $2 per pound and chicken costs $3 per pound. You have no more than $30 to spend. You expect to purchase at least 3 pounds of hamburgers.
- Write a system of inequalities to represent this situation.
- Graph the system of inequalities on the grid.
- Give three possible combinations for buying hamburgers and chicken for your summer bash.
Justify your answers
Answer:
A) System of inequalities:
- 2x + 3y ≤ 30
- x ≥ 3
- x ≥ 0
- y ≥ 0
B) Graph: see the picture attached
C) Three possible combinations:
- 3 pounds of hamburgers and 8 pounds of chicken
- 3 pounds of hamburgers and 0 pounds of chicken
- 15 pounds of hamburgers and 0 pounds of chicken
Explanation:
<u>A) Write the system of inequalities to represent this situaction.</u>
<u>1. Variables:</u>
- x: number of pound of hamburgers
- y: number of pound of chicken
<u>2. Costs:</u>
- x pounds of hamburgers at $2 per pound: 2x
- y: pounds of chicken at $3 per pound: 3y
- total cost: 2x + 3y
<u>3. First constraint:</u>
- You have no more than $ 30 to spend: means that the cost of what you buy can be at most (less than or equal to) $ 30.
- 2x + 3y ≤ 30
<u>4. Second constraint:</u>
- You expect to purchase at least 3 pounds of hamburgers: means that the number of pounds of hamburgers may be greater than or equal to 3.
- x ≥ 3
<u>5. Additional constraints:</u>
- Both, x and y cannot be negative: x, y ≥ 0
<u>6. System of equations:</u>
- 2x + 3y ≤ 30
- x ≥ 3
- x ≥ 0
- y ≥ 0
<u>B) Graph </u>
You have to graph all the constraints in a x-y coordinate system.
<u>1. To graph 2x + 3y ≤ 30 graph the line 2x + 3y = 30</u>
- Choose the y-intercept and x-intercept.
- x = 0 ⇒ 3y = 30 ⇒ y = 10 ⇒ point (0, 10)
- y = 0 ⇒ 2x =30 ⇒ x = 15 ⇒ point (15, 0)
- With two points you can draw the line
- Clear y: y ≤ 10 - 2x/3. Since, the symbol is ≤ you shade the region below the line, and the line is included, so you draw it as as solid line.
<u>2. To graph x ≥ 3 just draw the vertical line x = 3 and shade the region to the right of it. The points of the line are included (solid line).</u>
<u>3. The constraints x ≥ 0 and y ≥ 0 </u>mean that the region is restricted to the first quadrant (including the positive axis).
<u>4. The feasible solutions</u> are the set of points inside the common regions (intersection).
With all that information the graph is the one attached. The feasible solutions is the region defined by the triangle with vertices (3,8), (3,0), and (15,0).
<u>C) Give 3 possible combinations.</u>
You can pick any three points inside the region, as long as the coordinates are integer numbers. For instance the 3 vertices are solutions:
- (3, 8) ⇒ 3 pounds of hamburgers and 8 pounds of chicken
- (3,0) ⇒ 3 pounds of hamburgers and 0 pounds of chicken
- (15,0) ⇒ 15 pounds of hamburgers and 0 pounds of chicken
You could also prove that (5,4), 5 pounds of hamburgers and 4 pounds of chicken meet, the inequalities.
This is how you prove it:
- 5 ≥ 0
- 5 ≥ 3
- 4 ≥ 0
- 2(5) + 3(4) = 10 + 12 = 22 ≤ 30
And you can do the same for any pairs to verify whether they are solution or not.
