DSQ: Calculating x or y intercept - GMAT Quantitative Reasoning
Card 0 of 19
What is the 
-intercept of a line with equation 
-
-
What is the
The 
-intercept is the point at which 
. To find the 
-coordinate of the 
-intercept, just substitute 0 for 
:
Therefore, you need only know 
; knowing 
is neither necessary nor helpful.
If you are given that since 
. the 
-intercept is easily determined to be 
.
The answer is that Statement 2 alone is sufficient, but not Statement 1 alone.
The
Therefore, you need only know
If you are given that since
The answer is that Statement 2 alone is sufficient, but not Statement 1 alone.
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Find the 
-intercept and y-intercept of the following straight line.
1. The line has a slope of 0.6.
2. The line passes through point (10,2)
Find the
1. The line has a slope of 0.6.
2. The line passes through point (10,2)
To find the 
- and y-intercepets, we need both statements of information.
Statement 1 is not enough information because we don't have a reference point for the slope. A slope by itself is not enough to define the line. There are an infinite amount of lines with a slope of 0.6.
Statement 2 is not enough information by itself since it only tells us about 1 point on the line. Again, there are an infinite number of lines that pass through the point (10,2).
Only by using both statements can we find the 
- and y-intercepts. Solving, we see the line is actually
To find the
Statement 1 is not enough information because we don't have a reference point for the slope. A slope by itself is not enough to define the line. There are an infinite amount of lines with a slope of 0.6.
Statement 2 is not enough information by itself since it only tells us about 1 point on the line. Again, there are an infinite number of lines that pass through the point (10,2).
Only by using both statements can we find the
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Give the 
-intercept of the graph of the equation 
.
Statement 1: 
Statement 2: 
Give the
Statement 1:
Statement 2:
The 
-intercept of the graph of an equation is the point at which 
, so we evaluate 
:
The 
-intercept is simply the point 
, so knowing 
is necessary, and knowing 
is neither necessary nor helpful.
The
The
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Give the 
-intercept of the graph of the function 
Statement 1: 
Statement 2: 
Give the
Statement 1:
Statement 2:
To find the 
-intercept of 
, evaluate 
:
Knowing 
is necessary and sufficient; the value of 
is irrelevant.
To find the
Knowing
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Give the 
-intercept of the graph of the function 
Statement 1: 
Statement 2: 
Give the
Statement 1:
Statement 2:
To find the 
-intercept of 
, evaluate 
:
Knowing both 
and 
is necessary and sufficient.
To find the
Knowing both
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A line on the coordinate plane is neither horizontal nor vertical. Give its 
-intercept.
Statement 1: The line passes through 
.
Statement 2: The line passes through 
.
A line on the coordinate plane is neither horizontal nor vertical. Give its
Statement 1: The line passes through
Statement 2: The line passes through
Two points are necessary and sufficient to define a line. Therefore, neither statement alone is sufficient to determine the line, but both are sufficient. Once the line is defined, the 
-intercept - the point at which the line intersects the 
-axis - can be determined.
Two points are necessary and sufficient to define a line. Therefore, neither statement alone is sufficient to determine the line, but both are sufficient. Once the line is defined, the
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A function 
is graphed on the coordinate plane. Give the 
-intercept of the graph.
Statement 1: 
Statement 2: 
A function
Statement 1:
Statement 2:
The 
-intercept of the graph of 
is the point at which it intersects the 
-axis. Since this point has 
-coordinate 0, the 
-coordinate is 
. Statement 1 does not give us this value, but Statement 2 does.
The
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Continuous function 
has the set of all real numbers as its domain.
How many 
-intercepts does the graph 
have?
Statement 1: If 
, then 
.
Statement 2: 
.
Continuous function
How many
Statement 1: If
Statement 2:
The two statements together prvide insufficient information.
Assume both statements are true. By Statement 1, 
is a constantly increasing function, so it can intersect the 
-axis at most one time.
Now examine these two cases.
Case 1:
.
Also, if 
, then 
, so 
.
Since 
, the function has exactly one 
-intercept.
Case 2:
Also, if 
, then 
, so 
.
However, 2 raised to any power must be positive, so there is no value 
for which 
. The function has no 
-intercepts.
The two statements together prvide insufficient information.
Assume both statements are true. By Statement 1,
Now examine these two cases.
Case 1:
Also, if
Since
Case 2:
Also, if
However, 2 raised to any power must be positive, so there is no value
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Continuous function 
has the set of all real numbers as its domain.
How many 
-intercepts does the graph 
have?
Statement 1: If 
, then 
.
Statement 2: 
.
Continuous function
How many
Statement 1: If
Statement 2:
Statement 1 alone establishes that 
is always increasing. Its graph cannot have more than one 
-intercept; if it does, then the graph of the function must have a vertex between two intercepts, violating this statement. But it does not answer the question as to how many intercepts 
has, as seen in these two cases:
Case 1:
This is a linear function that is always increasing—it is in slope-intercept form, and its slope is 1, a positive number. The graph of 
has exactly one 
-intercept.
Case 2:
An exponential function with a base greater than 1, such as this, is an increasing function; however, 2 raised to any power must be positive, so there is no value 
for which 
. The graph of 
has no 
-intercepts.
Statement 2 alone establishes that at least one 
-intercept exists - since 
, 
is an 
-intercept. It does not, however, rule out the possibility of more 
-intercepts.
Now assume both statements are true. Since Statement 1 establishes that there is at most one 
-intercept, and Statement establishes that there is at least one 
-intercept, the two statements together establish that there is exactly one.
Statement 1 alone establishes that
Case 1:
This is a linear function that is always increasing—it is in slope-intercept form, and its slope is 1, a positive number. The graph of
Case 2:
An exponential function with a base greater than 1, such as this, is an increasing function; however, 2 raised to any power must be positive, so there is no value
Statement 2 alone establishes that at least one
Now assume both statements are true. Since Statement 1 establishes that there is at most one
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Continuous function 
has the set of all real numbers as its domain.
How many 
-intercepts does the graph 
have?
Statement 1: If 
, then 
.
Statement 2: If 
, then 
.
Continuous function
How many
Statement 1: If
Statement 2: If
Assume both statements are true.
By Statement 1, 
is decreasing on the domain interval 
; by Statement 2, 
is increasing on the domain interval 
. Therefore, 
must have its minimum value when 
.
This does not, however, tell us the number of 
-intercepts. For example, the graph of 
has as its minimum point 
, and, subsequently, exactly one 
-intercept. The graph of 
has as its minimum point 
and, subsequently, no 
-intercepts.
Assume both statements are true.
By Statement 1,
This does not, however, tell us the number of
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A function 
is graphed on the coordinate plane. It has exactly one 
-intercept. What is it?
Statement 1: 
Statement 2: 
A function
Statement 1:
Statement 2:
The 
-intercept of the graph of 
is the point at which it intersects the 
-axis. Since this point has 
-coordinate 0, the 
-coordinate is the value for which 
. Statement 1 gives us this value; Statement 2 does not.
The
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A line on the coordinate plane is neither horizontal nor vertical. Give its 
-intercept.
Statement 1: The line has slope 
.
Statement 1: The line passes through the origin.
A line on the coordinate plane is neither horizontal nor vertical. Give its
Statement 1: The line has slope
Statement 1: The line passes through the origin.
Statement 1 provides insufficient information, since the slope of the line alone is not enough from which to deduce the 
-intercept. Statement 2 alone tells us that the line passes through the point 
; since this is on the 
-axis, this is the 
-intercept.
Statement 1 provides insufficient information, since the slope of the line alone is not enough from which to deduce the
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and 
are two distinct nonvertical lines on the coordinate plane.
True or false: 
and 
have the same 
-intercept.
Statement 1: 
and 
intersect at 
.
Statement 2: 
and 
are perpendicular.
True or false:
Statement 1:
Statement 2:
The question is equivalent to asking whether the lines intersect at a point on the 
-axis.
Assume Statement 1 alone. Since 
and 
are distinct lines, 
, their common 
-intercept, is their sole point of intersection. They cannot intersect at a second point, so they cannot have the same 
-intercept.
Assume Statement 2 alone. Perpendicular lines are lines that meet at right angles; the question of their point of intersection is not answered by this statement.
The question is equivalent to asking whether the lines intersect at a point on the
Assume Statement 1 alone. Since
Assume Statement 2 alone. Perpendicular lines are lines that meet at right angles; the question of their point of intersection is not answered by this statement.
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and 
are two distinct nonvertical lines on the coordinate plane.
True or false: 
and 
have the same 
-intercept.
Statement 1: 
and 
have different 
-intercepts.
Statement 2: 
and 
both have slope 
.
True or false:
Statement 1:
Statement 2:
The question is equivalent to asking whether the lines intersect at a point on the 
-axis.
Statement 1 only establishes that the lines pass through different points on the 
-axis; no clues are given about any of their other points.
Statement 2 establishes that the lines have the same slope, and, subsequently, are parallel - that is, they do not intersect at all. Therefore, they cannot have the same 
-intercept.
The question is equivalent to asking whether the lines intersect at a point on the
Statement 1 only establishes that the lines pass through different points on the
Statement 2 establishes that the lines have the same slope, and, subsequently, are parallel - that is, they do not intersect at all. Therefore, they cannot have the same
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Stanley's teacher challenged him to write numbers in the circle, the square, and the triangle in the pattern below in order to make an equation whose graph has 
-intercept 
.
Did Stanley succeed?
Statement 1: Stanley wrote a 4 in the square.
Statement 2: Stanley wrote a 16 in the triangle.
Stanley's teacher challenged him to write numbers in the circle, the square, and the triangle in the pattern below in order to make an equation whose graph has
Did Stanley succeed?
Statement 1: Stanley wrote a 4 in the square.
Statement 2: Stanley wrote a 16 in the triangle.
Let 
stand for the values Stanley wrote in the square, the circle, and the triangle, respectively. The equation becomes
.
To find the 
-coordinate of the 
-intercept, set 
and solve for 
:
It is therefore necessary and sufficient to know the values of 
and 
- the values Ralph wrote in the square and the triangle - in order to determine the 
-intercept of Ralph's equation. Each statement alone give only one of those numbers; the two together give both.
Let
To find the
It is therefore necessary and sufficient to know the values of
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Ralph's teacher challenged him to write numbers in the circle, the square, and the triangle in the pattern below in order to make an equation whose graph has 
-intercept 
.
Did Ralph succeed?
Statement 1: Ralph wrote a 
in the square.
Statement 2: Ralph wrote a 
in the triangle.
Ralph's teacher challenged him to write numbers in the circle, the square, and the triangle in the pattern below in order to make an equation whose graph has
Did Ralph succeed?
Statement 1: Ralph wrote a
Statement 2: Ralph wrote a
Let 
stand for the values Ralph wrote in the square, the circle, and the triangle, respectively. The equation becomes
.
To find the 
-coordinate of the 
-intercept, set 
and solve for 
:
It is therefore necessary and sufficient to know the values of 
and 
- the values Ralph wrote in the circle and the triangle - in order to determine the 
-intercept of Ralph's equation. Statement 1 is irrelevant, and Statement 2 only provides the value Ralph wrote in the triangle.
Let
To find the
It is therefore necessary and sufficient to know the values of
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Gloria's teacher challenged her to write numbers in the circle and the square in the pattern below in order to make an equation whose graph has 
-intercept 
.
Did Gloria succeed?
Statement 1: Gloria wrote a 
in the square.
Statement 2: Gloria wrote a 
in the circle.
Gloria's teacher challenged her to write numbers in the circle and the square in the pattern below in order to make an equation whose graph has
Did Gloria succeed?
Statement 1: Gloria wrote a
Statement 2: Gloria wrote a
If we let 
stand for the number Gloria wrote in the square and 
stand for the number she wrote in the circle, the equation becomes
,
the slope-intercept form of the equation of a line. We can find the 
-coordinate of the 
-intercept by setting 
and solving for 
:
Therefore, it is necessary and sufficient to know both 
and 
- both of the numbers Gloria wrote - to determine the 
-intercept of the equation she made. Neither statement provides both values, but both statements together do.
If we let
the slope-intercept form of the equation of a line. We can find the
Therefore, it is necessary and sufficient to know both
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Ray's teacher challenged him to write numbers in the circle and the square in the pattern below in order to make an equation whose graph has 
-intercept 
.
Did Ray succeed?
Statement 1: Ray wrote a 5 in the square.
Statement 2: Ray wrote a 7 in the circle.
Ray's teacher challenged him to write numbers in the circle and the square in the pattern below in order to make an equation whose graph has
Did Ray succeed?
Statement 1: Ray wrote a 5 in the square.
Statement 2: Ray wrote a 7 in the circle.
If we let 
stand for the number Ray wrote in the square and 
stand for the number he wrote in the circle, the equation becomes
,
the slope-intercept form of the equation of a line. In this form, the 
-intercept is solely determined by the value of 
—namely, the value that Ray wrote in the circle. Statement 1 is therefore irrelevant, and Statement 2 alone establishes that Ray did not succeed.
If we let
the slope-intercept form of the equation of a line. In this form, the
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Mandy's teacher challenged her to write numbers in the circle, the square, and the triangle in the pattern below in order to make an equation whose graph has 
-intercept 
and 
-intercept 
.
Did Mandy succeed?
Statement 1: Mandy wrote a 5 in the triangle.
Statement 2: The number Mandy wrote in the square was five times the number she wrote in the circle.
Mandy's teacher challenged her to write numbers in the circle, the square, and the triangle in the pattern below in order to make an equation whose graph has
Did Mandy succeed?
Statement 1: Mandy wrote a 5 in the triangle.
Statement 2: The number Mandy wrote in the square was five times the number she wrote in the circle.
Let 
stand for the values Mandy wrote in the square, the circle, and the triangle, respectively. The equation becomes
.
From Statement 1 alone, we know that Mandy wrote a 5 in the triangle, but we do not know any of the others. The question of Mandy's success is unresolved.
Now assume Statement 2 alone is known. Then
.
The equation can be rewritten as
and it can be rewritten in slope-intercept form as
Mandy wrote an equation whose line has slope 
.
However, the slope of a line through 
and 
is
.
Statement 2 alone is sufficient to establish that Mandy did not succeed.
Let
From Statement 1 alone, we know that Mandy wrote a 5 in the triangle, but we do not know any of the others. The question of Mandy's success is unresolved.
Now assume Statement 2 alone is known. Then
The equation can be rewritten as
and it can be rewritten in slope-intercept form as
Mandy wrote an equation whose line has slope
However, the slope of a line through
Statement 2 alone is sufficient to establish that Mandy did not succeed.
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