A new coffee shop can hold no more than 50 seats. The owner wants at least 20 of the seats to be stools and the remaining seats to be recliners. If $x$ is the number of stools and $y$ is the number of recliners, which graph represents the solution to the system of inequalities?
[tex]
\begin{array}{l}
x+y \leq 50 \\
x \geq 20
\end{array}
[/tex]
Answer (2)
The problem provides two inequalities: x + y ≤ 50 and x ≥ 20 .
The first inequality can be rewritten as y ≤ − x + 50 , representing the region below the line y = − x + 50 .
The second inequality, x ≥ 20 , represents the region to the right of the vertical line x = 20 .
The solution is the region that satisfies both inequalities, bounded by x = 20 , y = 0 , and y = − x + 50 .
The intersection point is at x = 20 and y = 30 .
Explanation
Understanding the Problem We are given a system of inequalities that represent the constraints on the number of stools ( x ) and recliners ( y ) in a coffee shop. The inequalities are:
x + y ≤ 50 x ≥ 20
We need to determine which graph represents the solution to this system.
Analyzing the First Inequality First, let's analyze the first inequality, x + y ≤ 50 . This can be rewritten as y ≤ − x + 50 . This represents a region below the line y = − x + 50 .
Analyzing the Second Inequality Next, let's analyze the second inequality, x ≥ 20 . This represents a region to the right of the vertical line x = 20 .
Finding the Solution Region The solution to the system of inequalities is the region where both inequalities are satisfied. This is the region that is below the line y = − x + 50 and to the right of the line x = 20 . Also, since x and y represent the number of seats, they must be non-negative.
Determining the Boundaries To find the intersection of the two inequalities, we can substitute x = 20 into the first inequality:
20 + y ≤ 50 y ≤ 30
So, the region is bounded by x = 20 , y = 0 , y = − x + 50 , and y = 30 .
Examples
Imagine you're designing a community garden. You have a limited space (like the coffee shop's seat limit) and certain requirements (like the minimum number of stools). Inequalities help you determine the possible combinations of different plants (like stools and recliners) you can grow while staying within your constraints. This ensures you meet your minimum requirements and don't exceed your space.
The inequalities x + y ≤ 50 and x ≥ 20 define a solution region bounded by the line for the maximum total seats and the minimum number of stools. The feasible seating combination must be in a region below the line y = − x + 50 and to the right of x = 20 . This results in a triangular area representing valid combinations of stools and recliners.
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