Lines - GMAT Quantitative
Card 1 of 603
Given 
, find the equation of 
, a line 
to 
.
I) 
.
II) The 
-intercept of 
is at 
.
Given
I)
II) The
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To find the equation of a perpendicular line you need the slope of the line and a point on the line. We can find the slope by knowing g(x).
I) Gives us a point on h(x).
II) Gives us the y-intercept of h(x).
Either of these will be sufficient to find the rest of our equation.
To find the equation of a perpendicular line you need the slope of the line and a point on the line. We can find the slope by knowing g(x).
I) Gives us a point on h(x).
II) Gives us the y-intercept of h(x).
Either of these will be sufficient to find the rest of our equation.
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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
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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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Line m is perpendicular to the line l which is defined by the equation 
. What is the value of 
?
(1) Line m passes through the point 
.
(2) Line l passes through the point 
.
Line m is perpendicular to the line l which is defined by the equation
(1) Line m passes through the point
(2) Line l passes through the point
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Statement 1 allows you to define the equation of line m, but does not provide enough information to solve for 
. There are still 3 variables 
and only two different equations to solve.
if 
, statement 2 supplies enough information to solve for b by substitution if 
is on the line.
Statement 1 allows you to define the equation of line m, but does not provide enough information to solve for
if
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Consider linear functions 
and 
.
I) 
at the point 
.
II) 
Is the point 
on the line 
?
Consider linear functions
I)
II)
Is the point
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Consider linear functions h(t) and g(t).
I) 
at the point 
II) 
Is the point 
on the line h(t)?
We can use II) and I) to find the slope of h(t)
Recall that perpendicular lines have opposite reciprocal slope. Thus, the slope of h(t) must be 
Next, we know that h(t) must pass through (6,4), so lets us that to find the y-intercept:
Next, check if (10,4) is on h(t) by plugging it in.
So, the point is not on the line, and we needed both statements to know.
Consider linear functions h(t) and g(t).
I)
II)
Is the point
We can use II) and I) to find the slope of h(t)
Recall that perpendicular lines have opposite reciprocal slope. Thus, the slope of h(t) must be
Next, we know that h(t) must pass through (6,4), so lets us that to find the y-intercept:
Next, check if (10,4) is on h(t) by plugging it in.
So, the point is not on the line, and we needed both statements to know.
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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.
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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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Line AB is perpindicular to Line BC. Find the equation for Line AB.
1. Point B (the intersection of these two lines) is (2,5).
2. Line BC is parallel to the line y=2x.
Line AB is perpindicular to Line BC. Find the equation for Line AB.
1. Point B (the intersection of these two lines) is (2,5).
2. Line BC is parallel to the line y=2x.
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To find the equation of any line, we need 2 pieces of information, the slope of the line and any point on the line. From statement 1, we get a point on Line AB. From statement 2, we get the slope of Line BC. Since we know that AB is perpindicular to BC, we can derive the slope of AB from the slope of BC. Therefore to find the equation of the line, we need the information from both statements.
To find the equation of any line, we need 2 pieces of information, the slope of the line and any point on the line. From statement 1, we get a point on Line AB. From statement 2, we get the slope of Line BC. Since we know that AB is perpindicular to BC, we can derive the slope of AB from the slope of BC. Therefore to find the equation of the line, we need the information from both statements.
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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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Determine whether the points are collinear.
Statement 1: The three points are 
Statement 2: Slope of line 
and the slope of line 
Determine whether the points are collinear.
Statement 1: The three points are
Statement 2: Slope of line
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Points are collinear if they lie on the same line. Here A, B, and C are collinear if the line AB is the same as the line AC. In other words, the slopes of line AB and line AC must be the same. Statement 2 gives us the two slopes, so we know that Statement 2 is sufficient. Statement 1 also gives us all of the information we need, however, because we can easily find the slopes from the vertices. Therefore both statements alone are sufficient.
Points are collinear if they lie on the same line. Here A, B, and C are collinear if the line AB is the same as the line AC. In other words, the slopes of line AB and line AC must be the same. Statement 2 gives us the two slopes, so we know that Statement 2 is sufficient. Statement 1 also gives us all of the information we need, however, because we can easily find the slopes from the vertices. Therefore both statements alone are sufficient.
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Given:
Find 
.
I) 
.
II) The 
coordinate of the minmum of 
is 
.
Given:
Find
I)
II) The
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By using I) we know that the given point is on the line of the equation.
So I) is sufficient.
II) gives us the y coordinate of the minimum. In a quadratic equation, this is what "c" represents.
Therefore, c=-80 and II) is also sufficient.
By using I) we know that the given point is on the line of the equation.
So I) is sufficient.
II) gives us the y coordinate of the minimum. In a quadratic equation, this is what "c" represents.
Therefore, c=-80 and II) is also sufficient.
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Find whether the point 
is on the line 
.
I) 
is modeled by the following: 
.
II) 
is equal to five more than 3 times the y-intercept of 
.
Find whether the point
I)
II)
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To find out if a point is on line with an equation, one can simply plug in the point; however, this is complicated here by the fact that we are missing the x-coordinate.
Statement I gives us our function.
Statement II gives us a clue to find the value of 
. 
is five more than 3 times the y-intercept of 
. So, we can find the following:
To see if the point 
is on the line 
, plug it into the function:
This is not a true statement, so the point is not on the line.
To find out if a point is on line with an equation, one can simply plug in the point; however, this is complicated here by the fact that we are missing the x-coordinate.
Statement I gives us our function.
Statement II gives us a clue to find the value of
To see if the point
This is not a true statement, so the point is not on the line.
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A line segment has an endpoint at 
; what is its length?
-
Its other endpoint is 
-
Its midpoint is 
A line segment has an endpoint at
-
Its other endpoint is
-
Its midpoint is
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Given the other endpoint, you can use the distance formula to find the length of the segment:
Given the midpoint, you can use the distance formula to find the distance from the first endpoint to the midpoint, then double that to get the length of the segment:
The total length is twice that, or 10.
The answer is that either statement alone is sufficient to answer the question.
Given the other endpoint, you can use the distance formula to find the length of the segment:
Given the midpoint, you can use the distance formula to find the distance from the first endpoint to the midpoint, then double that to get the length of the segment:
The total length is twice that, or 10.
The answer is that either statement alone is sufficient to answer the question.
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Note: Figure NOT drawn to scale.
Give 
.
Statement 1: 
Statement 2: 
Note: Figure NOT drawn to scale.
Give
Statement 1:
Statement 2:
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If you know only that 
, then you know that 
and 
, but you still need 
, or a way finding it.
If you know only that 
, you still know only that 
, but you don't know their actual lengths.
If you know both facts, then you know 
If you know only that
If you know only that
If you know both facts, then you know
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Consider segment 
.
I) Point 
can be found at the point 
.
II) Segment 
had a length of 
units.
Find the coordinates of point 
.
Consider segment
I) Point
II) Segment
Find the coordinates of point
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Statement I gives us a point.
Statement II gives us the length of the segment.
We are asked to find the coordinates of the other end of the segment. However, we will need more information. Even with all of our information, we have no clue as to the orientation of the line. It could be 14 units straight up and down, it could be a perfectly horizontal line, or something inbetween, thus our answer is:
Neither I nor II are sufficient to answer the question. More information is needed.
Statement I gives us a point.
Statement II gives us the length of the segment.
We are asked to find the coordinates of the other end of the segment. However, we will need more information. Even with all of our information, we have no clue as to the orientation of the line. It could be 14 units straight up and down, it could be a perfectly horizontal line, or something inbetween, thus our answer is:
Neither I nor II are sufficient to answer the question. More information is needed.
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Find the length of Segment YZ
I) Point Y is located at the point 
.
II) Point Z has a y-coordinate twice that of Point Y and an x-coordinate one-third of Point Y.
Find the length of Segment YZ
I) Point Y is located at the point
II) Point Z has a y-coordinate twice that of Point Y and an x-coordinate one-third of Point Y.
Tap to reveal answer
To find the length of a segment, use the distance formula. The distance formula is given by the following:
Where your 
's and 
's correspond to the coordinates of the endpoints.
To find the length of Segment YZ, we need the endpoints.
Statement I gives you Point Y's coordinates.
Statement II relates Point Z's coordinates to Point Y's coordinates. Thus, we can find the point Z using Statement II.
Therefore, we need both.
Recap:
Find the length of Segment YZ
I) Point Y is located at the point 
.
II) Point Z has a y-coordinate twice that of Point Y, and an x-coordinate one-third of Point Y.
Use Statement II along with Statement I to find the coordinates of Point Z:
Then, use distance formula to find the length of Segment YZ:
To find the length of a segment, use the distance formula. The distance formula is given by the following:
Where your
To find the length of Segment YZ, we need the endpoints.
Statement I gives you Point Y's coordinates.
Statement II relates Point Z's coordinates to Point Y's coordinates. Thus, we can find the point Z using Statement II.
Therefore, we need both.
Recap:
Find the length of Segment YZ
I) Point Y is located at the point
II) Point Z has a y-coordinate twice that of Point Y, and an x-coordinate one-third of Point Y.
Use Statement II along with Statement I to find the coordinates of Point Z:
Then, use distance formula to find the length of Segment YZ:
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Consider segment 
I) Endpoint 
is located at the point 
.
Ii) Endpoint 
has an x-coordinate twice that of 
and a y-coordinate 15 times that of H.
What is the length of 
?
Consider segment
I) Endpoint
Ii) Endpoint
What is the length of
Tap to reveal answer
To find the length of a segment, we need both endpoints.
Statement I gives us one endpoint.
Statement II relates 
and 
, allowing us to find the second endpoint.
Thus, we need both. Once we have both endpoints, distance is easily calculated via the distance formula or the Pythagorean theorem.
Using Statement II, we find the second endpoint to be 
. Use the distance formula to find your answer:
To find the length of a segment, we need both endpoints.
Statement I gives us one endpoint.
Statement II relates
Thus, we need both. Once we have both endpoints, distance is easily calculated via the distance formula or the Pythagorean theorem.
Using Statement II, we find the second endpoint to be
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Find the equation to a line perpendicular to line 
.
- The slope of line
is 
.
- Line
goes through point 
.
Find the equation to a line perpendicular to line
- The slope of line
- Line
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Statement 1: Since the line we're looking for is perpendicular to line XY, our slope will be the inverse of line XY's slope 
.
The slope of our line is then 
. Just knowing the slope however, is not sufficient information to answer the question.
Statement 2: We're provided with a point which will allow us the write the equation.
Statement 1: Since the line we're looking for is perpendicular to line XY, our slope will be the inverse of line XY's slope
The slope of our line is then
Statement 2: We're provided with a point which will allow us the write the equation.
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Calculate the equation of a line perpendicular to line 
.
- The equation for line
is 
.
- Line
goes through point 
.
Calculate the equation of a line perpendicular to line
- The equation for line
- Line
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Statement 1: We're given the equation to line AB which contains the slope. Because the line we're being asked for is perpendicular to it, we know the slope will be its inverse.
The slope of our line is then 
Statement 2: We can write the equation to the perpendicular line only if we have a point that falls within that line. Luckily, we're given such a point in statement 2.
Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.
Statement 1: We're given the equation to line AB which contains the slope. Because the line we're being asked for is perpendicular to it, we know the slope will be its inverse.
The slope of our line is then
Statement 2: We can write the equation to the perpendicular line only if we have a point that falls within that line. Luckily, we're given such a point in statement 2.
Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.
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Find the equation of the line perpendicular to 
.
I) 
has a slope of 
.
II) The line must pass through the point 
.
Find the equation of the line perpendicular to
I)
II) The line must pass through the point
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Find the equation of the line perpendicular to r(x)
I) r(x) has a slope of -15
II) The line must pass through the point (9, 96)
Recall that perpendicular lines have opposite reciprocal slopes.
Use I) to find the slope of our new line
Use II) along with our slope to find the y-intercept of our new line.
Therefore both statements are needed.
Find the equation of the line perpendicular to r(x)
I) r(x) has a slope of -15
II) The line must pass through the point (9, 96)
Recall that perpendicular lines have opposite reciprocal slopes.
Use I) to find the slope of our new line
Use II) along with our slope to find the y-intercept of our new line.
Therefore both statements are needed.
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