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12 Nov 2014, 10:09
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D
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Difficulty:
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79% (01:14) correct
21%
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Tough and Tricky questions: Coordinate Geometry.
In the xy-coordinate plane, line \(l\) passes through the point \((-3, 0)\). Does line \(l\) pass through the point \((0, \frac{3}{4})\)?
(1) The slope of line \(l\) is \(\frac{1}{4}\).
(2) Line \(l\) passes through the point \((5, 2)\).
Kudos for a correct solution.
13 Nov 2014, 09:14
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Bunuel
Official Solution:
We start by writing the equation of line \(l\) in slope-intercept form: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. We are asked if the line passes through \((0, \frac{3}{4})\). Recall that the y-intercept is the point where the line crosses the y-axis, or, equivalently, the point on the line where \(x = 0\). Therefore, \((0, \frac{3}{4})\) is a potential y-intercept. We must determine if, in the equation of line \(l\), \(b\) is equal to \(\frac{3}{4}\).
Statement 1 gives us the slope of the line, which we can plug into the equation: \(y = \frac{1}{4}x + b\). In the prompt, we are also given a point on the line: \((-3, 0)\). Plugging in this point makes the equation: \(0 = \frac{1}{4}(-3) + b\). This simplifies to \(b = \frac{3}{4}\), and so line \(l\) does pass through the point \((0, \frac{3}{4})\). Statement 1 is sufficient to answer the question. Eliminate answer choices B, C, and E. The correct answer choice must be A or D.
Statement 2 tells us that line \(l\) passes through the point \((5, 2)\). Along with the x-intercept, \((-3, 0)\), we now have two points on the line, which is enough to find the slope. As we saw with statement 1, knowing the slope and one point on the line is enough to find the y-intercept. Since we can find the y-intercept, we know that we can find the answer to the question, even though we do not yet know whether the answer is yes or no. Therefore, statement 2 is also sufficient to answer the question.
Answer choice D is correct.
Recall that when we are given two points, we can always find the slope: \(slope = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}\). In this case, since our two points are \((-3, 0)\) and \((5, 2)\), \(m = \frac{0 - 2}{-3 - 5} = \frac{-2}{-8} = \frac{1}{4}\). We saw in examining statement 1 that if line \(l\) has slope equal to \(\frac{1}{4}\) and passes through the point \((-3, 0)\), it will have a y-intercept of \(\frac{3}{4}\), as desired.
Answer: D.
revul
Joined: 07 Oct 2014
Last visit: 07 May 2016
Posts: 160
Given Kudos: 17
Schools: Duke '17 (M) Tuck '17 (D) Ross '17 (A) Darden '17 (A) Cambridge'16 (WD) Kenan-Flagler '17 (A)
GMAT 1: 710 Q48 V39
WE:Sales (Other)
Schools: Duke '17 (M) Tuck '17 (D) Ross '17 (A) Darden '17 (A) Cambridge'16 (WD) Kenan-Flagler '17 (A)
GMAT 1: 710 Q48 V39
Posts: 160
12 Nov 2014, 10:19
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In the xy-coordinate plane, line l passes through the point (-3, 0). Does line l pass through the point (0, \frac{3}{4})?
(1) The slope of line l is \frac{1}{4}.
(2) Line l passes through the point (5, 2).
1). With this slope, the line increases height by 1/4 for each movement towards a higher X value. -3 to 0 is 3 movements. 3*1/4 = 3/4. sufficient.
2) delta y over delta x is slope. (2-0) / (5-(-3)) = 2/8 = 1/4 = same as 1) = sufficient.
Answer is D?
(1) The slope of line l is \frac{1}{4}.
(2) Line l passes through the point (5, 2).
1). With this slope, the line increases height by 1/4 for each movement towards a higher X value. -3 to 0 is 3 movements. 3*1/4 = 3/4. sufficient.
2) delta y over delta x is slope. (2-0) / (5-(-3)) = 2/8 = 1/4 = same as 1) = sufficient.
Answer is D?
