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:





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.

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.

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