DSQ: Calculating the slope of a line - GMAT Quantitative
Card 1 of 140
You are given two lines. Are they perpendicular?
Statement 1: The sum of their slopes is 
.
Statement 2: They have the same slope.
You are given two lines. Are they perpendicular?
Statement 1: The sum of their slopes is
Statement 2: They have the same slope.
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Statement 2 alone tells us that the lines are parallel, not perpendicular. Statement 1 alone is neither necessary nor helpful, as the sum of the slopes is irrelevant.
Statement 2 alone tells us that the lines are parallel, not perpendicular. Statement 1 alone is neither necessary nor helpful, as the sum of the slopes is irrelevant.
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Does a given line with intercepts 
have positive slope or negative slope?
Statement 1: 
Statement 2: 
Does a given line with intercepts
Statement 1:
Statement 2:
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The slope of a line through 
and 
is
If 
and 
have the same sign, then 
, making the slope negative; if 
and 
have the same sign, then 
, making the slope positive.
If we know both statements, we try to solve the system of equations as follows:
This means that the system is dependent, and that the statements are essentially the same.
Case 1: 
Then 
Case 2: 
Then 
Thus from Statement 1 alone, it cannot be determined whether and have the same sign, and the sign of the slope cannot be determined. Since Statement 2 is equivalent to Statement 1, the same holds of this statement, as well as both statements together.
The slope of a line through
If
If we know both statements, we try to solve the system of equations as follows:
This means that the system is dependent, and that the statements are essentially the same.
Case 1:
Then
Case 2:
Then
Thus from Statement 1 alone, it cannot be determined whether
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Is the slope of the line 
positve, negative, zero, or undefined?
Statement 1: 
Statement 2: 
Is the slope of the line
Statement 1:
Statement 2:
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, in slope-intercept form, is
Therefore, the sign of 
is the sign of the slope.
The first statement means that 
is positive - all that means is that both 
and 
are nonzero and of like sign. 
can be either positive or negative, and consequently, so can slope 
.
The second statement - that 
is positive - makes 
, the sign of the slope, negative.
Therefore, the sign of
The first statement means that
The second statement - that
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Does a given line with intercepts have positive slope or negative slope?
Statement 1: 
Statement 2: 
Does a given line with intercepts
Statement 1:
Statement 2:
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The slope of a line through and is
From Statement 1 alone, we can tell that
,
so we know the sign of the slope.
From Statement 2 alone, we can tell that
But this can be positive or negative - for example:
but
The slope of a line through
From Statement 1 alone, we can tell that
so we know the sign of the slope.
From Statement 2 alone, we can tell that
But this can be positive or negative - for example:
but
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Does a given line with intercepts have positive slope or negative slope?
Statement 1: 
Statement 2: 
Does a given line with intercepts
Statement 1:
Statement 2:
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The slope of a line through and is
If and have the same sign, then , making the slope negative; if and have the same sign, then , making the slope positive.
Statement 1 is not enough to determine the sign of 
.
Case 1: 
Case 2: 
So if we only know Statement 1, we do not know whether 
and 
have the same sign, and, subsequently, we do not know the sign of slope 
. A similar argument can be made that Statement 2 provides insufficient information.
If we know both statements, we can solve the system of equations as follows:
Therefore, we know 
and 
have unlike sign and 
.
The slope of a line through
If
Statement 1 is not enough to determine the sign of
Case 1:
Case 2:
So if we only know Statement 1, we do not know whether
If we know both statements, we can solve the system of equations as follows:
Therefore, we know
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A line includes points 
and 
. Is the slope of the line positive, negative, zero, or undefined?
Statement 1: 
Statement 2: 
A line includes points
Statement 1:
Statement 2:
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The slope of the line that includes points 
and 
is 
.
For the question of the sign of the slope to be answered, it must be known whether 
and 
are of the same sign or of different signs, or whether one of them is equal to zero.
Statement 1 alone does not answer this question, as it only states that the denominator is greater; it is possible for this to happen whether both are of like sign or unlike sign. Statement 2 only proves that 
- that is, that the denominator is positive.
If the two statements together are assumed, we know that 
. Since both the numerator and the denominator are positive, the slope of the line must be positive.
The slope of the line that includes points
For the question of the sign of the slope to be answered, it must be known whether
Statement 1 alone does not answer this question, as it only states that the denominator is greater; it is possible for this to happen whether both are of like sign or unlike sign. Statement 2 only proves that
If the two statements together are assumed, we know that
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A line is on the coordinate plane. What is its slope?
Statement 1: The line is parallel to the line of the equation 
.
Statement 2: The line is perpendicular to the line of the equation 
.
A line is on the coordinate plane. What is its slope?
Statement 1: The line is parallel to the line of the equation
Statement 2: The line is perpendicular to the line of the equation
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If Statement 1 alone holds - that is, if it is known only that the line is parallel to the line of 
- then this equation can be rewritten in slope-intercept form. The slope can be deduced from this, and the two lines, being parallel, will have the same slope.
If Statement 2 alone holds - that is, if it is known only that the line is perpendicular to the line of the equation 
- then this equation can be rewritten in slope-intercept form. The slope can be deduced from this, and the first line, which is perpendicular to this one, will have the slope that is the opposite of the reciprocal of that.
Either statement alone will yield an answer.
If Statement 1 alone holds - that is, if it is known only that the line is parallel to the line of
If Statement 2 alone holds - that is, if it is known only that the line is perpendicular to the line of the equation
Either statement alone will yield an answer.
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Is the slope of a line on the coordinate plane positive, zero, negative, or undefined?
Statement 1: The line contains points in both Quadrant I and Quadrant II.
Statement 2: The line contains points in both Quadrant I and Quadrant III.
Is the slope of a line on the coordinate plane positive, zero, negative, or undefined?
Statement 1: The line contains points in both Quadrant I and Quadrant II.
Statement 2: The line contains points in both Quadrant I and Quadrant III.
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Examine the diagram below.
It can be seen from the red lines that no conclusions about the sign of the slope of a line can be drawn from Statement 1, since lines of positive, negative, and zero slope can contain points in both Quadrant I and Quadrant II.
If a line contains a point in Quadrant I and a point in Quadrant III, then it contains a point with positive coordinates 
and a point with negative coordinates 
; its slope is
which is a positive slope.
Therefore, Statement 2 alone, but not Statement 1 alone, provides a definitive answer.
Examine the diagram below.
It can be seen from the red lines that no conclusions about the sign of the slope of a line can be drawn from Statement 1, since lines of positive, negative, and zero slope can contain points in both Quadrant I and Quadrant II.
If a line contains a point in Quadrant I and a point in Quadrant III, then it contains a point with positive coordinates
which is a positive slope.
Therefore, Statement 2 alone, but not Statement 1 alone, provides a definitive answer.
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Is the slope of a line on the coordinate plane positive, zero, negative, or undefined?
Statement 1: The line is perpendicular to the 
-axis.
Statement 2: The line has no 
-intercept.
Is the slope of a line on the coordinate plane positive, zero, negative, or undefined?
Statement 1: The line is perpendicular to the
Statement 2: The line has no
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The 
-axis is horizontal, so any line perpendicular to it is vertical and has undefined slope. Statement 1 is sufficient.
A line on the coordinate plane with no 
-intercept does not intersect the 
-axis and therefore must be parallel to it - subsequently, it must be vertical and have undefined slope. This makes Statement 2 sufficient.
The
A line on the coordinate plane with no
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Does a line on the coordinate plane have undefined slope?
Statement 1: It has 
-intercept 
Statement 2: It passes through Quadrant II.
Does a line on the coordinate plane have undefined slope?
Statement 1: It has
Statement 2: It passes through Quadrant II.
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A line with undefined slope is a vertical line.
Infinitely many lines, some vertical and some not, pass through 
, and infinitely many lines, some vertical and some not, pass through each quadrant, so neither statement alone is sufficient to answer the question.
Now assume both statements are true. Then, since the line passes through Quadrant II, it passes through at least one point with negative 
-coordinate and positive 
-coordinate, which we call 
. Its slope will be
,
a negative slope. Therefore, the slope is not undefined.
A line with undefined slope is a vertical line.
Infinitely many lines, some vertical and some not, pass through
Now assume both statements are true. Then, since the line passes through Quadrant II, it passes through at least one point with negative
a negative slope. Therefore, the slope is not undefined.
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Is the slope of a line on the coordinate plane positive, zero, negative, or undefined?
Statement 1: It includes the origin.
Statement 2: It passes through Quadrant II.
Is the slope of a line on the coordinate plane positive, zero, negative, or undefined?
Statement 1: It includes the origin.
Statement 2: It passes through Quadrant II.
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Infinitely many lines pass through the origin, and infinitely many lines pass through each quadrant, so neither statement alone is sufficient to answer the question.
Suppose that both statements are known to be true. Since the line passes through quadrant II, it passes through a point 
, where 
are positive. It also passes through 
so its slope will be
which is a negative slope.
Infinitely many lines pass through the origin, and infinitely many lines pass through each quadrant, so neither statement alone is sufficient to answer the question.
Suppose that both statements are known to be true. Since the line passes through quadrant II, it passes through a point
which is a negative slope.
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Is the slope of a line on the coordinate plane positive, zero, negative, or undefined?
Statement 1: It passes through the point 
.
Statement 2: It passes through Quadrant III.
Is the slope of a line on the coordinate plane positive, zero, negative, or undefined?
Statement 1: It passes through the point
Statement 2: It passes through Quadrant III.
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Examine the diagram below.
Both the red line and the green line fit the descriptions in both statements. The red line has negative slope and the green line has positive slope.
The two statements together give insufficient infomation.
Examine the diagram below.
Both the red line and the green line fit the descriptions in both statements. The red line has negative slope and the green line has positive slope.
The two statements together give insufficient infomation.
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Marisol was challenged by her teacher to fill in the square and the circle in the diagram below with two numbers to form the equation of a line with slope 
.
Did Marisol succeed?
Statement 1: Marisol wrote a 
in the box.
Statement 2: Marisol wrote a 
in the circle.
Marisol was challenged by her teacher to fill in the square and the circle in the diagram below with two numbers to form the equation of a line with slope
Did Marisol succeed?
Statement 1: Marisol wrote a
Statement 2: Marisol wrote a
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The pattern shows a linear equation in slope-intercept form, 
, with slope 
represented by the square and 
-intercept 
represented by the circle. Statement 1 gives the slope, so it provides sufficient information; Statement 2 gives only the 
-intercept, so it is unhelpful.
The pattern shows a linear equation in slope-intercept form,
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Tim was challenged by his teacher to fill in the square, the triangle, and the circle in the diagram below with three numbers to form the equation of a line with slope 
.
Did Tim succeed?
Statement 1: Tim wrote 
in the square.
Statement 2: Tim wrote 
in the circle.
Tim was challenged by his teacher to fill in the square, the triangle, and the circle in the diagram below with three numbers to form the equation of a line with slope
Did Tim succeed?
Statement 1: Tim wrote
Statement 2: Tim wrote
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Assume both statements are true. The equation takes the form
,
where 
is the number Tim wrote in the triangle.
Put the equation in slope-intercept form:
The coefficient of 
, which is 
, is the slope. As can be seen, the value of 
—that is, the number Tim wrote in the triangle—needs to be known. However, this information is still not given. Whether Tim succeeded is unknown.
Assume both statements are true. The equation takes the form
where
Put the equation in slope-intercept form:
The coefficient of
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Quinn was challenged by his teacher to fill in the square, the triangle, and the circle in the diagram below with three numbers to form the equation of a line with slope 
.
Did Quinn succeed?
Statement 1: Quinn wrote a 3 in the square.
Statement 2: Quinn wrote a 
in the triangle.
Quinn was challenged by his teacher to fill in the square, the triangle, and the circle in the diagram below with three numbers to form the equation of a line with slope
Did Quinn succeed?
Statement 1: Quinn wrote a 3 in the square.
Statement 2: Quinn wrote a
Tap to reveal answer
The equation is in standard form, with shapes replacing coefficients; that is, the equation is
.
Rewrite the equation in slope-intercept form:
The slope of the line of the equation is 
- and it depends on the number in the square, 
, and the triangle 
. Each statement alone gives the number in only one of the shapes; the two statements together give the numbers in both shapes and allow the slope to be calculated, thereby answering the question of Quinn's success.
The equation is in standard form, with shapes replacing coefficients; that is, the equation is
Rewrite the equation in slope-intercept form:
The slope of the line of the equation is
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Ava was challenged by her teacher to fill in the triangle, the square, and the circle in the diagram below with three numbers to form the equation of a line with slope 1.
Did Ava succeed?
Statement 1: Ava wrote a 1 in the circle.
Statement 2: Ava wrote the same positive number in both the triangle and the square.
Ava was challenged by her teacher to fill in the triangle, the square, and the circle in the diagram below with three numbers to form the equation of a line with slope 1.
Did Ava succeed?
Statement 1: Ava wrote a 1 in the circle.
Statement 2: Ava wrote the same positive number in both the triangle and the square.
Tap to reveal answer
Replacing the shapes with variables, the template becomes
Divide by 
to get the equation in slope-intercept form:
The slope is 
.
The slope is the ratio of the number in the square to the number in the triangle, so the number in the circle is irrelevant, making Statement 1 unhelpful.
Assume Statement 2 alone. Since the numbers in the square and the triangle are equal, 
, and 
. Ava succeeded.
Replacing the shapes with variables, the template becomes
Divide by
The slope is
The slope is the ratio of the number in the square to the number in the triangle, so the number in the circle is irrelevant, making Statement 1 unhelpful.
Assume Statement 2 alone. Since the numbers in the square and the triangle are equal,
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John's teacher gave him two equations, each with one coefficient missing, as follows:
John was challenged to write one number in each shape in order to form two equations whose lines have the same slope. Did he succeed?
Statement 1: The number John wrote in the box is three times the number he wrote in the circle.
Statement 2: John wrote 
in the circle.
John's teacher gave him two equations, each with one coefficient missing, as follows:
John was challenged to write one number in each shape in order to form two equations whose lines have the same slope. Did he succeed?
Statement 1: The number John wrote in the box is three times the number he wrote in the circle.
Statement 2: John wrote
Tap to reveal answer
Rewrite the two equations with variables rather than shapes:.
The first equation can be rewritten in slope-intercept form:
Its line has slope is 
, so if the number in the square is known, the slope is known.
is already in slope-intercept form; its line has slope 
, the number in the circle.
Statement 2 alone gives the number in the circle but provides no clue to the number in the square.
Now assume Statement 1 alone. Then 
. The slope of the first line is
,
the slope of the second line. Statement 1 provides sufficient proof that John was successful.
Rewrite the two equations with variables rather than shapes:.
The first equation can be rewritten in slope-intercept form:
Its line has slope is
is already in slope-intercept form; its line has slope
Statement 2 alone gives the number in the circle but provides no clue to the number in the square.
Now assume Statement 1 alone. Then
the slope of the second line. Statement 1 provides sufficient proof that John was successful.
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Veronica's teacher gave her two equations, the first with the coefficients of both variables missing, as follows:
Veronica was challenged to write one number in each shape in order to form an equation whose line has the same slope as that of the second equation. The only restriction was that she could not write a 5 in the square or a 3 in the circle.
Did Veronica write an equation with the correct slope?
Statement 1: Veronica wrote a negative integer in the square and a positive integer in the circle.
Statement 2: Veronica wrote an 8 in the circle.
Veronica's teacher gave her two equations, the first with the coefficients of both variables missing, as follows:
Veronica was challenged to write one number in each shape in order to form an equation whose line has the same slope as that of the second equation. The only restriction was that she could not write a 5 in the square or a 3 in the circle.
Did Veronica write an equation with the correct slope?
Statement 1: Veronica wrote a negative integer in the square and a positive integer in the circle.
Statement 2: Veronica wrote an 8 in the circle.
Tap to reveal answer
The slope of the line of
can be found by writing this equation in slope-intercept form:
The slope of the line is the coefficient of 
is 
, so Veronica must place the numbers in the shapes to yield an equation whose slope has this equation.
Rewrite the top equation as
The slope, in terms of 
and 
, can be found similarly:
Its slope is 
.
Statement 1 asserts that 
and 
are of unlike sign, so the slope 
must be negative. It cannot have sign 
, so the question is answered.
Assume Statement 2 alone. Then in the above equation, 
, so the slope is 
. The slope now depends on the value of 
, so Statement 2 gives insufficient information.
The slope of the line of
can be found by writing this equation in slope-intercept form:
The slope of the line is the coefficient of
Rewrite the top equation as
The slope, in terms of
Its slope is
Statement 1 asserts that
Assume Statement 2 alone. Then in the above equation,
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Give the slope of a line on the coordinate plane.
Statement 1: The line passes through the graph of the equation 
on the 
-axis.
Statement 2: The line passes through the graph of the equation 
on the 
-axis.
Give the slope of a line on the coordinate plane.
Statement 1: The line passes through the graph of the equation
Statement 2: The line passes through the graph of the equation
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The 
-intercept(s) of the graph of 
can be found by setting 
and solving for 
:
The graph has exactly one 
-intercept, 
.
The 
-intercept(s) of the graph of 
can be found by setting 
and solving for 
:
The graph has exactly one 
-intercept, 
.
In order to determine the slope of a line on the coordinate plane, the coordinates of two of its points are needed. Each of the two statements yields one of the points, so neither statement alone is sufficient to determine the slope. The two statements together, however, yield two points, and are therefore enough to determine the slope.
The
The graph has exactly one
The
The graph has exactly one
In order to determine the slope of a line on the coordinate plane, the coordinates of two of its points are needed. Each of the two statements yields one of the points, so neither statement alone is sufficient to determine the slope. The two statements together, however, yield two points, and are therefore enough to determine the slope.
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Give the slope of a line on the coordinate plane.
Statement 1: The line passes through the vertex of the parabola of the equation 
.
Statement 2: The line passes through the 
-intercept of the parabola of the equation 
.
Give the slope of a line on the coordinate plane.
Statement 1: The line passes through the vertex of the parabola of the equation
Statement 2: The line passes through the
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The vertex of the parabola of the equation 
can be found by first taking 
, then substituting in the equation and solving for 
.
The vertex is the point 
. Since 
, this is also the 
-intercept.
In order to determine the slope of a line on the coordinate plane, the coordinates of two of its points are needed. From the two statements together, we only know the 
-intercept and the vertex; however, they are one and the same. Therefore, we have insufficient information to find the slope.
The vertex of the parabola of the equation
The vertex is the point
In order to determine the slope of a line on the coordinate plane, the coordinates of two of its points are needed. From the two statements together, we only know the
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