Parallel Lines - GMAT Quantitative Reasoning
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Data Sufficiency Question
What is the slope of a line that passes through the point (2,3)?
1. It passes through the origin
2. It does not intersect with the line 
Data Sufficiency Question
What is the slope of a line that passes through the point (2,3)?
1. It passes through the origin
2. It does not intersect with the line
In order to calculate the equation of a line that passes through a point, we need one of two pieces of information. If we know another point, we can calculate the slope and solve for the 
-intercept, giving us the equation of the line. Alternatively, if we know the slope (which we can conclude from the parallel line in statement 2) we can calculate the 
-intercept and determine the equation of the line.
In order to calculate the equation of a line that passes through a point, we need one of two pieces of information. If we know another point, we can calculate the slope and solve for the
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Find the equation of the line parallel to the following line:
I) The new line passes through the point 
.
II) The new line has a 
-intercept of 
.
Find the equation of the line parallel to the following line:
I) The new line passes through the point
II) The new line has a
To find the equation of a parallel line, we need the slope and the y-intercept.
Parallel lines have the same slope, so we have that.
I and II each give us a point on the graph, so we could find the equation of the line through either of them.
To find the equation of a parallel line, we need the slope and the y-intercept.
Parallel lines have the same slope, so we have that.
I and II each give us a point on the graph, so we could find the equation of the line through either of them.
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Find the equation of the line 
.
- The slope of line
is 
.
- Line
goes through point 
.
Find the equation of the line
- The slope of line
- Line
Statement 1: We're given the slope line AB, because we are ask for the equation of the line we need more than just the slope of the line. Therefore, this information alone is not sufficient to write an actual equation.
Statement 2: Using the information from statement 1 and the points provided in this statement, we can answer the question.
Statement 1: We're given the slope line AB, because we are ask for the equation of the line we need more than just the slope of the line. Therefore, this information alone is not sufficient to write an actual equation.
Statement 2: Using the information from statement 1 and the points provided in this statement, we can answer the question.
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Given 
, find the equation of 
.
I) 
II) 
passes through the point 
Given
I)
II)
We are asked to find the equation of a line related to another line.
Statement I tells us the two lines are parallel. This means they have the same slope
Statement II gives us a point on our desired line. We can use this to find the line's y-intercept, which will then allow us to write its equation.
Plug all of the given info into slope-intercept form and solve for b, the line's y-intercept:
So our equation is:
We are asked to find the equation of a line related to another line.
Statement I tells us the two lines are parallel. This means they have the same slope
Statement II gives us a point on our desired line. We can use this to find the line's y-intercept, which will then allow us to write its equation.
Plug all of the given info into slope-intercept form and solve for b, the line's y-intercept:
So our equation is:
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Consider this system of equations:
Does this system has exactly one solution?
Statement 1: The lines representing the equations are parallel.
Statement 2: 
Consider this system of equations:
Does this system has exactly one solution?
Statement 1: The lines representing the equations are parallel.
Statement 2:
The solution of a system of linear equations is the point at which their lines intersect; if they are parallel, then by definition, there is no such point, and the system has no solution.
If 
, then we can rewrite the second equation as
In slope-intercept form:
This line has a slope of 
. The other equation has a line with slope of 
also, as can be easily seen since it is already in slope-intercept form. Since both equations have lines with the same slope, they are either the same line or parallel lines; either way, the system does not have exactly one solution.
The solution of a system of linear equations is the point at which their lines intersect; if they are parallel, then by definition, there is no such point, and the system has no solution.
If
In slope-intercept form:
This line has a slope of
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If 
is modeled by 
, find the slope of 
.
I) 
.
II) 
crosses the 
-axis at 
.
If
I)
II)
I) Tells us the two lines are parallel. Parallel lines have the same slope.
II) Gives us the x-intercept of b(t). By itself this gives us no clue as to the slop of b. If we had another point on b(t) we could find the slope, but we don't have another point.
So, statement I is what we need.
I) Tells us the two lines are parallel. Parallel lines have the same slope.
II) Gives us the x-intercept of b(t). By itself this gives us no clue as to the slop of b. If we had another point on b(t) we could find the slope, but we don't have another point.
So, statement I is what we need.
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Calculate the slope of a line parallel to line 
.
- Line
passes through points 
and 
.
- The equation for line
is 
.
Calculate the slope of a line parallel to line
- Line
- The equation for line
Statement 1: Since we're referring to a line parallel to line XY, the slopes will be identical. We can use the points provided to calculate the slope:
We can simplify the slope to just 
.
Statement 2: Finding the slope of a line parallel to line XY is really straightforward when given the equation of a line.
Where 
is the slope and 
the y-intercept.
In this case, our 
value is 
.\
Each statement alone is sufficient to answer the question.
Statement 1: Since we're referring to a line parallel to line XY, the slopes will be identical. We can use the points provided to calculate the slope:
We can simplify the slope to just
Statement 2: Finding the slope of a line parallel to line XY is really straightforward when given the equation of a line.
Where
In this case, our
Each statement alone is sufficient to answer the question.
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Find the slope of the line parallel to 
.
I) 
passes through the point 
.
II) 
has an x-intercept of 290.
Find the slope of the line parallel to
I)
II)
Recall that parallel lines have the same slope and that slope can be calculated from any two points.
Statement I gives us a point on 
Statement II gives us the x-intercept, a.k.a. the point 
.
Therfore, using both statements, we can find the slope of 
and any line parallel to it.
Recall that parallel lines have the same slope and that slope can be calculated from any two points.
Statement I gives us a point on
Statement II gives us the x-intercept, a.k.a. the point
Therfore, using both statements, we can find the slope of
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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
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.
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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.
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.
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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
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.
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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
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,
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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
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
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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
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.
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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
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.
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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
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.
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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
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.
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