unit review a: linear systems
problem 4 a fast food restaurant employs cooks and cashiers. cooks earn $500 per week and cashiers earn $250 per week. the restaurant employs at most 15 employees at a given time. the restaurant’s weekly payroll for cooks and cashiers cannot exceed $5,000.
a. write a system inequalities to represent the possible employment combinations.
b. represent the system of inequalities on a graph.
graph with x-axis: no. of cooks (0 - 30), y-axis: no. of cashiers (0 - 30)
c. prepare two possible employment scenarios (solutions) that satisfy the constraints of this situation.
problem 5 a local radio station hosts a pool party where they promote their station by giving out free concert tickets and baseball caps. they are permitted to distribute up to 100 items and they expect to give away at least 10 concert tickets and 50 baseball caps.
write a system of inequalities representing the possible combinations of concert tickets and baseball caps given away at the radio station’s pool party.
problem 6 graph the system of linear inequalities.
\begin{cases}
x \geq -3 \\
y 1 \\
y x \\
y \leq -\frac{1}{3}x + 4
\end{cases}
graph with x and y axes, grid
© math beach solutions llc

Question

unit review a: linear systems
problem 4 a fast food restaurant employs cooks and cashiers. cooks earn $500 per week and cashiers earn $250 per week. the restaurant employs at most 15 employees at a given time. the restaurant’s weekly payroll for cooks and cashiers cannot exceed $5,000.
a. write a system inequalities to represent the possible employment combinations.
b. represent the system of inequalities on a graph.
graph with x-axis: no. of cooks (0 - 30), y-axis: no. of cashiers (0 - 30)
c. prepare two possible employment scenarios (solutions) that satisfy the constraints of this situation.
problem 5 a local radio station hosts a pool party where they promote their station by giving out free concert tickets and baseball caps. they are permitted to distribute up to 100 items and they expect to give away at least 10 concert tickets and 50 baseball caps.
write a system of inequalities representing the possible combinations of concert tickets and baseball caps given away at the radio station’s pool party.
problem 6 graph the system of linear inequalities.
\begin{cases}
 x \geq -3 \\
 y > 1 \\
 y > x \\
 y \leq -\frac{1}{3}x + 4
\end{cases}
graph with x and y axes, grid
© math beach solutions llc

Accepted Answer

### Problem 4A
# Explanation:
## Step1: Define Variables
Let $ x $ = number of cooks, $ y $ = number of cashiers.
## Step2: Employee Limit Inequality
Total employees: $ x + y \leq 15 $ (at most 15 employees).
## Step3: Payroll Limit Inequality
Payroll: $ 500x + 250y \leq 5000 $. Simplify: divide by 250, get $ 2x + y \leq 20 $.
## Step4: Non - negative Constraints
$ x \geq 0 $, $ y \geq 0 $ (can't have negative employees).

# Answer:
The system of inequalities is $\begin{cases}x + y\leq15\2x + y\leq20\x\geq0\y\geq0\end{cases}$

### Problem 4B
# Explanation:
## Step1: Graph $ x + y = 15 $
Find intercepts: when $ x = 0 $, $ y = 15 $; when $ y = 0 $, $ x = 15 $. Draw solid line (since $ \leq $), shade below.
## Step2: Graph $ 2x + y = 20 $
Intercepts: $ x = 0 $, $ y = 20 $; $ y = 0 $, $ x = 10 $. Solid line, shade below.
## Step3: Graph $ x\geq0,y\geq0 $
Shade in first quadrant (where $ x$ and $ y$ are non - negative).
## Step4: Find Feasible Region
The feasible region is the intersection of all shaded areas (below both lines, in first quadrant).

# Answer:
The graph has the $ x$ - axis (cooks) and $ y$ - axis (cashiers). The line $ x + y = 15 $ connects (0,15) and (15,0), line $ 2x + y = 20 $ connects (0,20) and (10,0). The feasible region is the polygon bounded by $ x = 0 $, $ y = 0 $, $ x + y = 15 $, and $ 2x + y = 20 $ (the overlapping area in the first quadrant below both lines).

### Problem 4C
# Explanation:
## Step1: Find First Scenario
Let's pick $ x = 5 $ (cooks). Plug into $ x + y\leq15 $: $ 5 + y\leq15\Rightarrow y\leq10 $. Plug into $ 2x + y\leq20 $: $ 10 + y\leq20\Rightarrow y\leq10 $. So $ y = 10 $ (cashiers). Check payroll: $ 5\times500+10\times250 = 2500 + 2500 = 5000 $, which is within limit. Employees: $ 5 + 10 = 15 $, within limit.
## Step2: Find Second Scenario
Let $ x = 0 $ (cooks). Then from $ x + y\leq15 $, $ y\leq15 $; from $ 2x + y\leq20 $, $ y\leq20 $. So $ y = 15 $ (cashiers). Payroll: $ 0\times500+15\times250 = 3750\leq5000 $. Employees: $ 0 + 15 = 15 $, within limit.

# Answer:
Scenario 1: 5 cooks and 10 cashiers. Scenario 2: 0 cooks and 15 cashiers. (Other valid scenarios are also possible, e.g., 10 cooks and 0 cashiers: $ 10\times500+0\times250 = 5000 $, $ 10 + 0 = 10\leq15 $)

### Problem 5
# Explanation:
## Step1: Define Variables
Let $ x $ = number of concert tickets, $ y $ = number of baseball caps.
## Step2: Total Items Inequality
Total items: $ x + y\leq100 $ (up to 100 items).
## Step3: Ticket Minimum Inequality
$ x\geq10 $ (at least 10 tickets).
## Step4: Cap Minimum Inequality
$ y\geq50 $ (at least 50 caps).
## Step5: Non - negative Constraints (Implicit)
$ x\geq0 $, $ y\geq0 $ (but $ x\geq10 $ and $ y\geq50 $ are stricter here).

# Answer:
The system of inequalities is $\begin{cases}x + y\leq100\x\geq10\y\geq50\end{cases}$

### Problem 6
# Explanation:
## Step1: Graph $ x\geq - 3 $
Draw a vertical line $ x=-3 $, solid line (since $ \geq $), shade to the right of the line.
## Step2: Graph $ y > 1 $
Draw a horizontal dashed line $ y = 1 $ (since $ > $, not inclusive), shade above the line.
## Step3: Graph $ y > x $
Draw the line $ y = x $ (dashed, since $ > $), shade above the line.
## Step4: Graph $ y\leq-\frac{1}{3}x + 4 $
Find intercepts: $ x = 0 $, $ y = 4 $; $ y = 0 $, $ x = 12 $. Solid line (since $ \leq $), shade below the line.
## Step5: Find Feasible Region
The feasible region is the intersection of all shaded areas: right of $ x=-3 $, above $ y = 1 $, above $ y = x $, and below $ y=-\frac{1}{3}x + 4 $.

# Answer:
The graph has a vertical line $ x=-3 $ (shade right), horizontal dashed line $ y = 1 $ (shade above), line $ y = x $ (dashed, shade above), and line $ y=-\frac{1}{3}x + 4 $ (solid, shade below). The feasible region is the area that satisfies all these conditions (e.g., for $ x$ between - 3 and values where $ y = x $ and $ y=-\frac{1}{3}x + 4 $ intersect, above $ y = 1 $). To find intersection of $ y = x $ and $ y=-\frac{1}{3}x + 4 $: $ x=-\frac{1}{3}x + 4\Rightarrow\frac{4}{3}x = 4\Rightarrow x = 3 $, $ y = 3 $. So the feasible region is bounded by $ x=-3 $, $ y = 1 $, $ y = x $, and $ y=-\frac{1}{3}x + 4 $ in the appropriate area.

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QUESTION IMAGE

Question

Explanation:

Problem 4A

Step1: Define Variables

Step2: Employee Limit Inequality

Step3: Payroll Limit Inequality

Step4: Non - negative Constraints

Step1: Graph \( x + y = 15 \)

Step2: Graph \( 2x + y = 20 \)

Step3: Graph \( x\geq0,y\geq0 \)

Step4: Find Feasible Region

Step1: Find First Scenario

Step2: Find Second Scenario

Answer:

Problem 4B