DSQ: Calculating whether lines are parallel - GMAT Quantitative
Card 0 of 63
One line includes the points 
and 
; a second line includes the points 
and 
. If these lines are parallel, what is the value of 
?
-
-
One line includes the points
Tap to see back →
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 
, then you can easily find 
by substituting:
Cross-multiply and solve:
If you know that 
, do the same thing:
Therefore, 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.
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.
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.
Tap to see back →
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.
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.
You are given two distinct lines, 
and 
on the coordinate plane. Are they parallel lines, perpendicular lines, or neither of these?
Statement 1: The absolute value of the slope of Line 
is 1.
Statement 2: The absolute value of the slope of Line 
is 1.
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
Tap to see back →
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.
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.
You are given two distinct lines, Line 
and Line 
, on the coordinate plane. Neither line is horizontal or vertical. Are they parallel lines, perpendicular lines, or neither of these?
Statement 1: The product of the slopes of the two lines is 
.
Statement 2: The absolute value of the slope of Line 
is 
.
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
Tap to see back →
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 
are not, but in both cases, the product of the slopes is 1.
Statement 2 alone gives that Line 
has slope 1 or 
, but nothing is given about the slope of Line 
.
Now, assume both statements are true. From Statement 2, 
has slope 1 or 
. From Statement 1, the product of the slopes is 1; if the slope of 
is 1, then the slope of 
is 
, and if the slope of 
is 
, then the slope of 
is 
. Therefore, if both statements are true, the lines have the same slope, making them parallel.
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,
You are given distinct lines 
and 
on the coordinate plane. Are they parallel, perpendicular, or neither?
Statement 1: The product of the slopes of the two lines is 
.
Statement 2: The slope of Line 
is 
.
You are given distinct lines
Statement 1: The product of the slopes of the two lines is
Statement 2: The slope of Line
Tap to see back →
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 
. Two lines with slope 3 are parallel, and one line with slope 1 and one with slope 9 are neither parallel nor perpendicular; both pairs of lines satisfy Statement 1, but only the first pair is parallel. Therefore, Statement 1 only establishes that they are not perpendicular.
From Statement 2, only the slope of 
is given; without the slope of 
, the question cannot be answered.
Assume both statements to be true. Then since Line 
has slope 
and the product of the slopes is 9, The slope of Line 
is 
. Therefore, both lines have slope 
, and the lines are parallel.
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
You are given distinct lines 
and 
on the coordinate plane. Are they parallel, perpendicular, or neither?
Statement 1: Line 
has 
-intercept 
and line 
has 
-intercept 
.
Statement 1: Line 
has 
-intercept 
and line 
has 
-intercept 
.
You are given distinct lines
Statement 1: Line
Statement 1: Line
Tap to see back →
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.
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.
You are given distinct lines 
and 
on the coordinate plane. Are they parallel, perpendicular, or neither?
Statement 1: Line 
has slope 3 and Line 
has slope 
.
Statement 2: Line 
has 
-intercept 
and Line 
has 
-intercept 
.
You are given distinct lines
Statement 1: Line
Statement 2: Line
Tap to see back →
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 
, the lines are perpendicular.
Statement 2 alone is unhelpful, since no information about the slope of a line can be determined from only one point.
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.
You are given distinct lines 
and 
on the coordinate plane. Are they parallel, perpendicular, or neither?
Statement 1: Line 
has slope 3 and Line 
has slope 
.
Statement 2: Both lines have 
-intercept 
.
You are given distinct lines
Statement 1: Line
Statement 2: Both lines have
Tap to see back →
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 
; since the product of the slopes is 
, this choice can be eliminated as well. It can therefore be deduced that the lines are neither parallel nor perpendicular.
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.
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.
You are given distinct lines 
and 
on the coordinate plane. Are they parallel, perpendicular, or neither?
Statement 1: Both lines have slope 3.
Statement 2: Line 
has 
-intercept 
and Line 
has 
-intercept 
.
You are given distinct lines
Statement 1: Both lines have slope 3.
Statement 2: Line
Tap to see back →
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.
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.
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.
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.
Tap to see back →
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.
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.
One line includes the points and ; a second line includes the points and . If these lines are parallel, what is the value of ?
-
-
One line includes the points
Tap to see back →
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 , then you can easily find by substituting:
Cross-multiply and solve:
If you know that , do the same thing:
Therefore, 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.
You are given two distinct lines, and on the coordinate plane. Are they parallel lines, perpendicular lines, or neither of these?
Statement 1: The absolute value of the slope of Line is 1.
Statement 2: The absolute value of the slope of Line is 1.
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
Tap to see back →
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.
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.
You are given two distinct lines, Line and Line , on the coordinate plane. Neither line is horizontal or vertical. Are they parallel lines, perpendicular lines, or neither of these?
Statement 1: The product of the slopes of the two lines is .
Statement 2: The absolute value of the slope of Line is .
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
Tap to see back →
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 are not, but in both cases, the product of the slopes is 1.
Statement 2 alone gives that Line has slope 1 or , but nothing is given about the slope of Line .
Now, assume both statements are true. From Statement 2, has slope 1 or . From Statement 1, the product of the slopes is 1; if the slope of is 1, then the slope of is , and if the slope of is , then the slope of is . Therefore, if both statements are true, the lines have the same slope, making them parallel.
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,
You are given distinct lines and on the coordinate plane. Are they parallel, perpendicular, or neither?
Statement 1: The product of the slopes of the two lines is .
Statement 2: The slope of Line is .
You are given distinct lines
Statement 1: The product of the slopes of the two lines is
Statement 2: The slope of Line
Tap to see back →
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 . Two lines with slope 3 are parallel, and one line with slope 1 and one with slope 9 are neither parallel nor perpendicular; both pairs of lines satisfy Statement 1, but only the first pair is parallel. Therefore, Statement 1 only establishes that they are not perpendicular.
From Statement 2, only the slope of is given; without the slope of , the question cannot be answered.
Assume both statements to be true. Then since Line has slope and the product of the slopes is 9, The slope of Line is . Therefore, both lines have slope , and the lines are parallel.
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
You are given distinct lines and on the coordinate plane. Are they parallel, perpendicular, or neither?
Statement 1: Line has -intercept and line has -intercept .
Statement 1: Line has -intercept and line has -intercept .
You are given distinct lines
Statement 1: Line
Statement 1: Line
Tap to see back →
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.
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.
You are given distinct lines and on the coordinate plane. Are they parallel, perpendicular, or neither?
Statement 1: Line has slope 3 and Line has slope .
Statement 2: Line has -intercept and Line has -intercept .
You are given distinct lines
Statement 1: Line
Statement 2: Line
Tap to see back →
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 , the lines are perpendicular.
Statement 2 alone is unhelpful, since no information about the slope of a line can be determined from only one point.
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.
You are given distinct lines and on the coordinate plane. Are they parallel, perpendicular, or neither?
Statement 1: Line has slope 3 and Line has slope .
Statement 2: Both lines have -intercept .
You are given distinct lines
Statement 1: Line
Statement 2: Both lines have
Tap to see back →
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 ; since the product of the slopes is , this choice can be eliminated as well. It can therefore be deduced that the lines are neither parallel nor perpendicular.
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.
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.
You are given distinct lines and on the coordinate plane. Are they parallel, perpendicular, or neither?
Statement 1: Both lines have slope 3.
Statement 2: Line has -intercept and Line has -intercept .
You are given distinct lines
Statement 1: Both lines have slope 3.
Statement 2: Line
Tap to see back →
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.
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.
One line includes the points and ; a second line includes the points and . If these lines are parallel, what is the value of ?
-
-
One line includes the points
Tap to see back →
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 , then you can easily find by substituting:
Cross-multiply and solve:
If you know that , do the same thing:
Therefore, 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.
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.
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.
Tap to see back →
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.
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.