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Line ℓ lies in the xy-plane and does not pass through the origin. What is the slope of line ℓ ?
We need to find the slope of line \(\ell\) that lies in the \(xy\)-plane and does not pass through the origin.
Given Information:
- Line \(\ell\) lies in the \(xy\)-plane
- Line \(\ell\) does not pass through the origin \((0,0)\)
What We Need to Determine:
Can we find exactly ONE value for the slope of line \(\ell\)?
Key Insight: Since the line doesn't pass through the origin, it must have at least one intercept. To determine a unique slope, we need information that pins down the line to exactly one possibility.
Statement 1: The x-intercept of line \(\ell\) is twice the y-intercept of line \(\ell\).
Let's think about what this relationship means:
When a line has both intercepts, it passes through two specific points on the axes. Statement 1 tells us these intercepts have a fixed ratio - the x-intercept is exactly twice the y-intercept.
Visualize it this way: If the y-intercept is at height 3, then the x-intercept must be at distance 6. If the y-intercept is at height 5, then the x-intercept must be at distance 10. No matter what value you choose for one intercept, the other is locked in by this \(2:1\) ratio.
This fixed ratio is like giving someone precise directions: "For every 1 unit up from the origin, go 2 units across." There's only one angle, one slope, that satisfies this requirement.
[STOP - Sufficient!]
Statement 1 is sufficient.
This eliminates choices B, C, and E.
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2: The x- and y-intercepts of line \(\ell\) are both positive.
This tells us the line crosses both axes in the positive region, but consider the possibilities:
Example 1: A steep line crossing at \((1, 0)\) and \((0, 10)\) - slope would be very negative
Example 2: A gentle line crossing at \((10, 0)\) and \((0, 1)\) - slope would be slightly negative
Example 3: A moderate line crossing at \((5, 0)\) and \((0, 5)\) - slope would be \(-1\)
All three lines satisfy "both intercepts are positive," yet each has a different slope. In fact, we can draw infinitely many lines connecting any point on the positive y-axis to any point on the positive x-axis.
Statement 2 is NOT sufficient.
This eliminates choices B and D.
Statement 1 alone gives us a unique slope through the fixed ratio constraint, while Statement 2 alone allows infinitely many possible slopes.
Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."