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16 Sep 2014, 01:49
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D
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Difficulty:
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Question Stats:
82% (01:20) correct
18%
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wrong
based on 50
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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) The slope of line \(l\) is \(\frac{1}{4}\).
(2) Line \(l\) passes through the point \((5, 2)\).
16 Sep 2014, 01:49
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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
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
11 Oct 2014, 06:22
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Nice explanation Bunuel,
I took the quick intuitive approach. Since we know one point that the line passes through (does not matter where, in this instance), we can say with certainty whether the line will pass through any other given point as long as we have the slope. Stmt 1 gives the slope directly, so if we wanted to we could test for sure if a line with that slope will cross the point in question or not. But we don't need to- sufficient. Stmt 2 gives us another set of x,y coordinates from which we can calculate the slope using the points given in the question stem, so again, sufficient. Ans = D
I took the quick intuitive approach. Since we know one point that the line passes through (does not matter where, in this instance), we can say with certainty whether the line will pass through any other given point as long as we have the slope. Stmt 1 gives the slope directly, so if we wanted to we could test for sure if a line with that slope will cross the point in question or not. But we don't need to- sufficient. Stmt 2 gives us another set of x,y coordinates from which we can calculate the slope using the points given in the question stem, so again, sufficient. Ans = D
