GMAT Math : DSQ: Calculating whether lines are parallel
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Example Question #1 : Dsq: Calculating Whether Lines Are Parallel
You are given two lines. Are they parallel?
Statement 1: The product of their slopes is 
Statement 2: One has positive slope; one has negative slope.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Two parallel lines must have the same slope. Therefore, the product of the slopes will be the product of two real numbers of like sign, which must be positive. Each of the two statements contradicts this, so either statement alone answers the question.
Example Question #2 : Dsq: Calculating Whether Lines Are Parallel
One line includes the points 
1)
2)
BOTH statements TOGETHER are insufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is not sufficient.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is not sufficient.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
The lines are parallel, so their slopes are equal.
The slope of the first is 
The slope of the second is 
Set the two equal to each other:
If you know that 
Cross-multiply and solve:
If you know that 
Therefore, either statement alone is sufficient to answer the question.
Example Question #3 : Dsq: Calculating Whether Lines Are Parallel
You are given distinct lines 
Statement 1: Both lines have slope 3.
Statement 2: Line 
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Two lines can be determined to be parallel, perpendicular, or neither from their slopes.
Assume Statement 1 alone is true. Since these distinct lines have the same slope, they are parallel.
Assume Statement 2 alone is true. No information about the slopes of the lines can be determined from one single point, so Statement 2 alone is insufficient.
Example Question #4 : Dsq: Calculating Whether Lines Are Parallel
You are given distinct lines 
Statement 1: Line 
Statement 2: Both lines have 
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Two lines can be determined to be parallel, perpendicular, or neither from their slopes.
Assume Statement 1 alone. Parallel lines must have the same slope, so this choice can be eliminated. The slopes of perpendicular lines must have product 
Assume Statement 2 alone. Since the lines have at least one point in common, they are not parallel, but this is the only choice that can be eliminated.
Example Question #5 : Dsq: Calculating Whether Lines Are Parallel
You are given distinct lines 
Statement 1: Line 
Statement 2: Line 
BOTH statements TOGETHER are insufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Two lines are parallel if and only if they have the same slope, and perpendicular if and only if their slopes have product 
Assume Statement 1 alone. Since the product of the slopes is 
Statement 2 alone is unhelpful, since no information about the slope of a line can be determined from only one point.
Example Question #6 : Dsq: Calculating Whether Lines Are Parallel
You are given two distinct lines, 
Statement 1: The absolute value of the slope of Line 
Statement 2: The absolute value of the slope of Line 
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Assume both statements are true. Then three things are possible:
Case 1: Both lines will have slope 1, or
Case 2: Both lines will have slope
In either case, since the lines have the same slope, they are parallel.
Case 3: One line has slope 1 and one has slope
In this case the lines are perpendicular.
The two statements therefore provide insufficient information.
Example Question #7 : Dsq: Calculating Whether Lines Are Parallel
You are given distinct lines 
Statement 1: Line 
Statement 1: Line 
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
The answer to the question depends on the slopes of the lines—parallel lines have the same slope, and perpendicular lines have slopes that have product 
From Statement 1 alone, we only know one point of each line, so no information about their slopes can be determined; the same holds for Statement 2.
Assume both statements are true. Then we know two points of each line—specifically, both intercepts—from which we can determine the slope of each by way of the slope formula. After doing so, we can use the slopes to answer the question.
Example Question #8 : Dsq: Calculating Whether Lines Are Parallel
You are given two distinct lines, Line 
Statement 1: The product of the slopes of the two lines is 
Statement 2: The absolute value of the slope of Line 
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
The question can be answered by finding and comparing the slopes of the lines. The lines are parallel if and only if they have the same slope, and perpendicular if and only if the product of the slopes is 
Statement 1 alone does not answer the question. Two lines with slope 1 are parallel, and a line with slope 2 and a line with slope 
Statement 2 alone gives that Line 
Now, assume both statements are true. From Statement 2, 
Example Question #9 : Dsq: Calculating Whether Lines Are Parallel
You are given distinct lines 
Statement 1: The product of the slopes of the two lines is 
Statement 2: The slope of Line 
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
The answer to the question depends on the slopes of the lines - parallel lines have the same slope, and perpendicular lines have slopes that have product 
Statement 1 alone only eliminates the possiblity of the lines being perpendicular, since the product of the slope is not 
From Statement 2, only the slope of 
Assume both statements to be true. Then since Line 
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