An artist creates custom oil paintings and charcoal sketches. * Each oil painting, x, requ… — SAT

An artist creates custom oil paintings and charcoal sketches.

* Each oil painting, $x$ , requires $4$ hours of setup time and $2$ hours of finishing time.
* Each charcoal sketch, $y$ , requires $1$ hour of setup time and $3$ hours of finishing time.

Each week, the artist can spend at most $20$ hours on setup time and at most $30$ hours on finishing time. Which of the following systems of inequalities represents this situation?

A
$\begin{aligned} 4x + y &\ge 20 \\ 2x + 3y &\ge 30 \end{aligned}$
B
$\begin{aligned} 4x + 2y &\le 20 \\ x + 3y &\le 30 \end{aligned}$
$\begin{aligned} 4x + y &\le 20 \\ 2x + 3y &\le 30 \end{aligned}$ Answer
D
$\begin{aligned} 4x + y &\le 30 \\ 2x + 3y &\le 20 \end{aligned}$

Answer

The system of inequalities consisting of the inequalities

4x + y \le 20

and

2x + 3y \le 30

The variable

x

represents the number of oil paintings and

y

represents the number of charcoal sketches. Setup time constraint: each oil painting requires

4

hours and each charcoal sketch requires

1

hour. Thus, the total setup time is

4x + y

hours. Since the artist can spend at most

20

hours on setup time, this is represented by the inequality

4x + y \le 20

. Finishing time constraint: each oil painting requires

2

hours and each charcoal sketch requires

3

hours. Thus, the total finishing time is

2x + 3y

hours. Since the artist can spend at most

30

hours on finishing time, this is represented by the inequality

2x + 3y \le 30

. Combining these gives the correct system.

Step-by-Step Solution

Identify the variables and setup times for each type of artwork to write the setup time inequality.

The total setup time is

4x + y

hours. Since the limit is at most

20

hours, the inequality is

4x + y \le 20

To establish the mathematical representation of the setup time constraint.

Identify the variables and finishing times for each type of artwork to write the finishing time inequality.

The total finishing time is

2x + 3y

hours. Since the limit is at most

30

hours, the inequality is

2x + 3y \le 30

To establish the mathematical representation of the finishing time constraint.

Combine the two inequalities into a system.

The system of inequalities is

4x + y \le 20

and

2x + 3y \le 30

To find the system that represents both constraints simultaneously.

Key Concept

Translating real-world constraints into a system of linear inequalities in two variables.

Question

Answer

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Key Concept

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