Can this be solved algebraically?

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Can this be solved algebraically?

by DHILLONRAVI1983 » Sat Nov 12, 2016 6:22 pm
OG 2017 Diagnostic Quant question. Can this be solved algebraically?

In the XY plane, if line k has negative slope and passes through the point (-5, r), is the x intercept of line k positive?

1. the slope of line k is -5
2. r>0

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by GMATGuruNY » Sat Nov 12, 2016 6:56 pm
DHILLONRAVI1983 wrote:In the XY plane, if line k has negative slope and passes through the point (-5, r), is the x intercept of line k positive?

1. the slope of line k is -5
2. r>0
Both statements are satisfied by the following cases:

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Since the x-intercept is POSITIVE in the first case but NEGATIVE in the second case, the two statements combined are INSUFFICIENT.

The correct answer is E.
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by GMATGuruNY » Sat Nov 12, 2016 7:11 pm
An algebraic way to combine the two statements:

The equation of a line is y = mx + b.
The x-intercept = -b/m.

Statement 1: m = -5
Thus:
y = -5x + b, with the result that the x-intercept = -b/m = -b/-5 = b/5.

Statement 2: In point (-5, r), r>0.
Substituting x = -5 and y = r into y = -5x + b, we get:
r = 25 + b.

Since r>0, we get:
25 + b > 0
b > - 25.

Case 1: b = -24, with the result that the x-intercept = b/5 = -24/5.
Case 2: b = 1, with the result that the x-intercept = b/5 = 1/5.

Since the x-intercept is NEGATIVE in Case 1 but POSITIVE in Case 2, the two statements combined are INSUFFICIENT.

The correct answer is E.
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by Jeff@TargetTestPrep » Wed Nov 16, 2016 8:00 am
DHILLONRAVI1983 wrote:OG 2017 Diagnostic Quant question. Can this be solved algebraically?

In the XY plane, if line k has negative slope and passes through the point (-5, r), is the x intercept of line k positive?

1. the slope of line k is -5
2. r>0
We need to determine whether the x-intercept of line k is positive, given that line k has a negative slope and passes through the point (-5, r).

We can let the slope of line k be m and the x-intercept of line k be a; that is, line k passes through the point (a, 0). Using the slope formula m = (y_2 - y_1)/(x_2 - x_1), we have:

m = (0 - r)/(a -(-5))= -r/(a + 5)

Statement One Alone:

The slope of line k is -5.

Using the information in statement one, we can say the following:

-r/(a + 5) = -5

a + 5 = r/5

a = r/5 - 5

Since we don't know the value of r, we cannot determine the value of a.

For example, if r = 5, a = -4, which is negative. However, if r = 50, a = 5, which is positive. Statement one alone is not sufficient to answer the question. We can eliminate answer choices A and D.

Statement Two Alone:

r > 0

Knowing r is positive does not give us enough information to determine whether a, the x-intercept of line k, is positive. Statement two alone is not sufficient to answer the question. We can eliminate answer choice B.

Statements One and Two Together:

Looking at our work from statement one, and keeping in mind that r > 0 from statement two, we see that a can be positive or negative.

For example, if r = 5 then a = -4, which is negative. However, if r = 50 then a = 5, which is positive. The two statements together are still not sufficient to answer the question.

Answer: E

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