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In the xy-plane, region R consists of all the points \(\mathrm{(x,y)}\) such that \(\mathrm{2x + 3y \leq 6}\). Is the point \(\mathrm{(r,s)}\) in region R?
Let's break down what we're being asked. Region R consists of all points \(\mathrm{(x,y)}\) where \(\mathrm{2x + 3y \leq 6}\). We need to determine if point \(\mathrm{(r,s)}\) is in this region.
What We Need to Determine: Is \(\mathrm{2r + 3s \leq 6}\)?
This is a yes/no question. For sufficiency, we need to definitively answer either YES (the point is in region R) or NO (the point is not in region R).
Key Insight: Region R includes everything on or below the line \(\mathrm{2x + 3y = 6}\). Think of this line as a boundary—points below it are in the region, points above it are not.
Statement 1 tells us: The point \(\mathrm{(r,s)}\) lies on the line \(\mathrm{3r + 2s = 6}\).
Here's the critical observation: We have a point that must lie on one specific line \(\mathrm{(3r + 2s = 6)}\), and we need to know if it's in a region defined by a different line \(\mathrm{(2r + 3s \leq 6)}\).
Since these two lines have different slopes, they must intersect at exactly one point. This means the line \(\mathrm{3r + 2s = 6}\) must cross through region R's boundary—it will have:
Let's verify with two quick examples:
Since different points on the line give different answers, we cannot determine whether \(\mathrm{(r,s)}\) is in region R.
Statement 1 is NOT sufficient. [STOP - Not Sufficient!]
This eliminates answer choices A and D.
Now let's forget Statement 1 completely and analyze Statement 2 by itself.
Statement 2 tells us: The point must satisfy \(\mathrm{r \leq 3}\) and \(\mathrm{s \leq 2}\).
This creates a rectangular region in the plane. To determine if this entire rectangle is inside region R, we can use a strategic approach: check the corners.
Why corners? Because with linear inequalities like \(\mathrm{2r + 3s \leq 6}\), the maximum and minimum values occur at the vertices of the region.
Testing the extreme points:
Since the rectangular region contains points both inside and outside region R, we cannot determine whether \(\mathrm{(r,s)}\) is in region R.
Statement 2 is NOT sufficient. [STOP - Not Sufficient!]
This eliminates answer choices B and D (already eliminated).
Combined Information: The point \(\mathrm{(r,s)}\) must:
1. Lie on the line \(\mathrm{3r + 2s = 6}\) (from Statement 1)
2. Satisfy \(\mathrm{r \leq 3}\) and \(\mathrm{s \leq 2}\) (from Statement 2)
This means we're looking at the intersection of a line with a rectangle—which gives us a line segment.
From Statement 1, we can express: \(\mathrm{s = (6 - 3r)/2 = 3 - 1.5r}\)
For this to also satisfy \(\mathrm{s \leq 2}\):
Combined with \(\mathrm{r \leq 3}\) from Statement 2, we get: \(\mathrm{2/3 \leq r \leq 3}\)
Now let's check the endpoints of this line segment:
Actually, when \(\mathrm{s = 0}\) (the bottom boundary of our rectangle): \(\mathrm{3r = 6}\), so \(\mathrm{r = 2}\)
So even with both statements combined, we have:
The statements together are NOT sufficient. [STOP - Not Sufficient!]
This eliminates answer choices A, B, C, and D.
Since we can find points satisfying both statements that are inside region R AND other points satisfying both statements that are outside region R, we cannot definitively answer whether \(\mathrm{(r,s)}\) is in region R.
Answer Choice E: "The statements together are not sufficient."