ssyohee wrote:1. If the y-intercept of a line is greater than 2, the x-intercept is greater than 3?
1) The line passes through (2, 3)
2) The slope of the line is 1/2

the answer is can anyone please explain why?
Thanks!

Say the y-intercept is (2 + p), where p is a positive number

Let's make use of intercept form of the Straight line.

The standard form is:

x/a + y/b = 1; where a and b are x-intercept and y-intercept, respectively

We are given that b = 2 + p

Thus,
x/a + y/(2+p) = 1

We have to ensure whether a > 3.

Statement 1: The line passes through (2, 3).

Thus,
2/a + 3/(2+p) = 1

Let's plug-in a = x-intercept = 3 and see whether p, a positive number, is still positive.

Thus,
2/3 + 3/(2+p) = 1
3/(2+p) = 1/3
2 + p = 9

p = 7, a positive number

=> x-intercept is NOT greater than 3.

Let's try with a > 3; say a = 4

Thus,
2/4 + 3/(2+p) = 1
3/(2+p) = 1/2
2 + p = 6

p = 4, a positive number

=> x-intercept is greater than 3. No unique answer. Insufficient!

Statement 2: The slope of the line is 1/2.

Let's make use of slope and intercept form of the staright line.

Say the equation of the line is: y = mx + c; where m is the slope and c is y-intercept.

Thus,
y = x/2 + (2+p)

Say x-intercept = a, thus, we deduce that the line passes through the point (a, 0).

Thus,
0 = a/2 + (2 + p)
0 = a + 4 + 2p
a = -(4 + 2p)

=> x-intercept = |a| = |-(4 + 2p)| = 4 + 2p

Since p is a positive number x-intercept = |a| > 4 or greater than 3. Suffcient.

The correct answer: B

Hope this helps!

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The line meets the y-axis above the origin. Using Statement 1, we know (2, 3) is on the line. Here it's best to draw different diagrams, placing the y-intercept above and below y=3, to see where the x-intercept will be in different cases. If the y-intercept is only very slightly greater than 2, ( say at y = 2.1), then the line will be rising as it moves to the right, and the x-intercept will then be negative (so will certainly be less than 3). If instead the y-intercept is slightly greater than 3, say at y=4, then the line will be falling as it moves to the right, and as long as the slope is not extremely steep, the x-intercept will be greater than 3. So Statement 1 is not sufficient.

Using Statement 2, if the y-intercept is positive, and the slope is positive, then the line is rising as it moves to the right, so the x-intercept will be negative, and certainly not greater than 3, so Statement 2 is sufficient, and the answer is B.

The question is strange, though, because the two statements together contradict the information in the question stem, which is something that can never happen in a real GMAT question. If (2, 3) is on a line, and that line has slope 1/2, then the y-intercept of that line is exactly 2. It cannot be greater than 2, which is what the stem says. So either the question is not well-designed, or there's possibly a negative sign missing somewhere (I think the question is more interesting if the slope in Statement 2 is -1/2, rather than +1/2). Where is the question from?