Line ℓ lies in the xy-plane and does not pass through the origin. What is the slope of line ℓ...

Understanding the Question

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.

Analyzing Statement 1

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.

Analyzing Statement 2

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.

The Answer: A

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."