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
Can this be solved algebraically?
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- DHILLONRAVI1983
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Both statements are satisfied by the following cases: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
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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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.
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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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).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 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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