Lines - GRE Quantitative Reasoning
Card 0 of 848
If the coordinates (3, 14) and (_–_5, 15) are on the same line, what is the equation of the line?
If the coordinates (3, 14) and (_–_5, 15) are on the same line, what is the equation of the line?
First solve for the slope of the line, m using y=mx+b
m = (y2 – y1) / (x2 – x1)
= (15 – 14) / (_–_5 _–_3)
= (1 )/( _–_8)
=_–_1/8
y = –(1/8)x + b
Now, choose one of the coordinates and solve for b:
14 = –(1/8)3 + b
14 = _–_3/8 + b
b = 14 + (3/8)
b = 14.375
y = –(1/8)x + 14.375
First solve for the slope of the line, m using y=mx+b
m = (y2 – y1) / (x2 – x1)
= (15 – 14) / (_–_5 _–_3)
= (1 )/( _–_8)
=_–_1/8
y = –(1/8)x + b
Now, choose one of the coordinates and solve for b:
14 = –(1/8)3 + b
14 = _–_3/8 + b
b = 14 + (3/8)
b = 14.375
y = –(1/8)x + 14.375
Compare your answer with the correct one above
What is the equation of the line passing through (–1,5) and the upper-right corner of a square with a center at the origin and a perimeter of 22?
What is the equation of the line passing through (–1,5) and the upper-right corner of a square with a center at the origin and a perimeter of 22?
If the square has a perimeter of 22, each side is 22/4 or 5.5. This means that the upper-right corner is (2.75, 2.75)—remember that each side will be "split in half" by the x and y axes.
Using the two points we have, we can ascertain our line's equation by using the point-slope formula. Let us first get our slope:
m = rise/run = (2.75 – 5)/(2.75 + 1) = –2.25/3.75 = –(9/4)/(15/4) = –9/15 = –3/5.
The point-slope form is: y – y0 = m(x – x0). Based on our data this is: y – 5 = (–3/5)(x + 1); Simplifying, we get: y = (–3/5)x – (3/5) + 5; y = (–3/5)x + 22/5
If the square has a perimeter of 22, each side is 22/4 or 5.5. This means that the upper-right corner is (2.75, 2.75)—remember that each side will be "split in half" by the x and y axes.
Using the two points we have, we can ascertain our line's equation by using the point-slope formula. Let us first get our slope:
m = rise/run = (2.75 – 5)/(2.75 + 1) = –2.25/3.75 = –(9/4)/(15/4) = –9/15 = –3/5.
The point-slope form is: y – y0 = m(x – x0). Based on our data this is: y – 5 = (–3/5)(x + 1); Simplifying, we get: y = (–3/5)x – (3/5) + 5; y = (–3/5)x + 22/5
Compare your answer with the correct one above
Which line passes through the points (0, 6) and (4, 0)?
Which line passes through the points (0, 6) and (4, 0)?
P1 (0, 6) and P2 (4, 0)
First, calculate the slope: m = rise ÷ run = (y2 – y1)/(x2 – x1), so m = –3/2
Second, plug the slope and one point into the slope-intercept formula:
y = mx + b, so 0 = –3/2(4) + b and b = 6
Thus, y = –3/2x + 6
P1 (0, 6) and P2 (4, 0)
First, calculate the slope: m = rise ÷ run = (y2 – y1)/(x2 – x1), so m = –3/2
Second, plug the slope and one point into the slope-intercept formula:
y = mx + b, so 0 = –3/2(4) + b and b = 6
Thus, y = –3/2x + 6
Compare your answer with the correct one above
What line goes through the points (1, 3) and (3, 6)?
What line goes through the points (1, 3) and (3, 6)?
If P1(1, 3) and P2(3, 6), then calculate the slope by m = rise/run = (y2 – y1)/(x2 – x1) = 3/2
Use the slope and one point to calculate the intercept using y = mx + b
Then convert the slope-intercept form into standard form.
If P1(1, 3) and P2(3, 6), then calculate the slope by m = rise/run = (y2 – y1)/(x2 – x1) = 3/2
Use the slope and one point to calculate the intercept using y = mx + b
Then convert the slope-intercept form into standard form.
Compare your answer with the correct one above
Let y = 3_x_ – 6.
At what point does the line above intersect the following:
Let y = 3_x_ – 6.
At what point does the line above intersect the following:
If we rearrange the second equation it is the same as the first equation. They are the same line.
If we rearrange the second equation it is the same as the first equation. They are the same line.
Compare your answer with the correct one above
What is the slope of the equation 4_x_ + 3_y_ = 7?
What is the slope of the equation 4_x_ + 3_y_ = 7?
We should put this equation in the form of y = mx + b, where m is the slope.
We start with 4_x_ + 3_y_ = 7.
Isolate the y term: 3_y_ = 7 – 4_x_
Divide by 3: y = 7/3 – 4/3 * x
Rearrange terms: y = –4/3 * x + 7/3, so the slope is –4/3.
We should put this equation in the form of y = mx + b, where m is the slope.
We start with 4_x_ + 3_y_ = 7.
Isolate the y term: 3_y_ = 7 – 4_x_
Divide by 3: y = 7/3 – 4/3 * x
Rearrange terms: y = –4/3 * x + 7/3, so the slope is –4/3.
Compare your answer with the correct one above
What is the slope of the line with equation 4_x_ – 16_y_ = 24?
What is the slope of the line with equation 4_x_ – 16_y_ = 24?
The equation of a line is:
y = mx + b, where m is the slope
4_x_ – 16_y_ = 24
–16_y_ = –4_x_ + 24
y = (–4_x_)/(–16) + 24/(–16)
y = (1/4)x – 1.5
Slope = 1/4
The equation of a line is:
y = mx + b, where m is the slope
4_x_ – 16_y_ = 24
–16_y_ = –4_x_ + 24
y = (–4_x_)/(–16) + 24/(–16)
y = (1/4)x – 1.5
Slope = 1/4
Compare your answer with the correct one above
Which of the following lines is parallel to:
Which of the following lines is parallel to:
First write the equation in slope intercept form. Add 
to both sides to get 
. Now divide both sides by 
to get 
. The slope of this line is 
, so any line that also has a slope of 
would be parallel to it. The correct answer is 
.
First write the equation in slope intercept form. Add
Compare your answer with the correct one above
There are two lines:
2x – 4y = 33
2x + 4y = 33
Are these lines perpendicular, parallel, non-perpendicular intersecting, or the same lines?
There are two lines:
2x – 4y = 33
2x + 4y = 33
Are these lines perpendicular, parallel, non-perpendicular intersecting, or the same lines?
To be totally clear, solve both lines in slope-intercept form:
2x – 4y = 33; –4y = 33 – 2x; y = –33/4 + 0.5x
2x + 4y = 33; 4y = 33 – 2x; y = 33/4 – 0.5x
These lines are definitely not the same. Nor are they parallel—their slopes differ. Likewise, they cannot be perpendicular (which would require not only opposite slope signs but also reciprocal slopes); therefore, they are non-perpendicular intersecting.
To be totally clear, solve both lines in slope-intercept form:
2x – 4y = 33; –4y = 33 – 2x; y = –33/4 + 0.5x
2x + 4y = 33; 4y = 33 – 2x; y = 33/4 – 0.5x
These lines are definitely not the same. Nor are they parallel—their slopes differ. Likewise, they cannot be perpendicular (which would require not only opposite slope signs but also reciprocal slopes); therefore, they are non-perpendicular intersecting.
Compare your answer with the correct one above
Which pair of linear equations represent parallel lines?
Which pair of linear equations represent parallel lines?
Parallel lines will always have equal slopes. The slope can be found quickly by observing the equation in slope-intercept form and seeing which number falls in the "
" spot in the linear equation 
,
We are looking for an answer choice in which both equations have the same value. Both lines in the correct answer have a slope of 2, therefore they are parallel.
Parallel lines will always have equal slopes. The slope can be found quickly by observing the equation in slope-intercept form and seeing which number falls in the "
We are looking for an answer choice in which both equations have the same
Compare your answer with the correct one above
Which of the following equations represents a line that is parallel to the line represented by the equation 
?
Which of the following equations represents a line that is parallel to the line represented by the equation
Lines are parallel when their slopes are the same.
First, we need to place the given equation in the slope-intercept form.
Because the given line has the slope of 
, the line parallel to it must also have the same slope.
Lines are parallel when their slopes are the same.
First, we need to place the given equation in the slope-intercept form.
Because the given line has the slope of
Compare your answer with the correct one above
Which of the above-listed lines are parallel?
Which of the above-listed lines are parallel?
There are several ways to solve this problem. You could solve all of the equations for 
. This would give you equations in the form 
. All of the lines with the same 
value would be parallel. Otherwise, you could figure out the ratio of 
to 
when both values are on the same side of the equation. This would suffice for determining the relationship between the two. We will take the first path, though, as this is most likely to be familiar to you.
Let's solve each for 
:
Here, you need to be a bit more manipulative with your equation. Multiply the numerator and denominator of the 
value by 
:
Therefore, 
, 
, and 
all have slopes of 
There are several ways to solve this problem. You could solve all of the equations for
Let's solve each for
Here, you need to be a bit more manipulative with your equation. Multiply the numerator and denominator of the
Therefore,
Compare your answer with the correct one above
Which of the following is parallel to the line passing through 
and 
?
Which of the following is parallel to the line passing through
Now, notice that the slope of the line that you have been given is 
. You know this because slope is merely:
However, for your points, there is no rise at all. You do not even need to compute the value. You know it will be 
. All lines with slope 
are of the form 
, where 
is the value that 
has for all 
points. Based on our data, this is 
, for 
is always 
—no matter what is the value for 
. So, the parallel answer choice is 
, as both have slopes of 
.
Now, notice that the slope of the line that you have been given is
However, for your points, there is no rise at all. You do not even need to compute the value. You know it will be
Compare your answer with the correct one above
Which of the following is parallel to 
?
Which of the following is parallel to
To begin, solve your equation for 
. This will put it into slope-intercept form, which will easily make the slope apparent. (Remember, slope-intercept form is 
, where 
is the slope.)
Divide both sides by 
and you get:
Therefore, the slope is 
. Now, you need to test your points to see which set of points has a slope of 
. Remember, for two points 
and 
, you find the slope by using the equation:
For our question, the pair 
and 
gives us a slope of 
:
To begin, solve your equation for
Divide both sides by
Therefore, the slope is
For our question, the pair
Compare your answer with the correct one above
If the line through the points (5, –3) and (–2, p) is parallel to the line y = –2_x_ – 3, what is the value of p ?
If the line through the points (5, –3) and (–2, p) is parallel to the line y = –2_x_ – 3, what is the value of p ?
Since the lines are parallel, the slopes must be the same. Therefore, (p+3) divided by (_–2–_5) must equal _–_2. 11 is the only choice that makes that equation true. This can be solved by setting up the equation and solving for p, or by plugging in the other answer choices for p.
Since the lines are parallel, the slopes must be the same. Therefore, (p+3) divided by (_–2–_5) must equal _–_2. 11 is the only choice that makes that equation true. This can be solved by setting up the equation and solving for p, or by plugging in the other answer choices for p.
Compare your answer with the correct one above
What line is parallel to 
and passes through the point 
?
What line is parallel to
Start by converting the original equation to slop-intercept form.
The slope of this line is 
. A parallel line will have the same slope. Now that we know the slope of our new line, we can use slope-intercept form and the given point to solve for the y-intercept.
Plug the y-intercept into the slope-intercept equation to get the final answer.
Start by converting the original equation to slop-intercept form.
The slope of this line is
Plug the y-intercept into the slope-intercept equation to get the final answer.
Compare your answer with the correct one above
There is a line defined by the equation below:
There is a second line that passes through the point 
and is parallel to the line given above. What is the equation of this second line?
There is a line defined by the equation below:
There is a second line that passes through the point
Parallel lines have the same slope. Solve for the slope in the first line by converting the equation to slope-intercept form.
3x + 4y = 12
4y = _–_3x + 12
y = –(3/4)x + 3
slope = _–_3/4
We know that the second line will also have a slope of _–_3/4, and we are given the point (1,2). We can set up an equation in slope-intercept form and use these values to solve for the y-intercept.
y = mx + b
2 = _–_3/4(1) + b
2 = _–_3/4 + b
b = 2 + 3/4 = 2.75
Plug the y-intercept back into the equation to get our final answer.
y = –(3/4)x + 2.75
Parallel lines have the same slope. Solve for the slope in the first line by converting the equation to slope-intercept form.
3x + 4y = 12
4y = _–_3x + 12
y = –(3/4)x + 3
slope = _–_3/4
We know that the second line will also have a slope of _–_3/4, and we are given the point (1,2). We can set up an equation in slope-intercept form and use these values to solve for the y-intercept.
y = mx + b
2 = _–_3/4(1) + b
2 = _–_3/4 + b
b = 2 + 3/4 = 2.75
Plug the y-intercept back into the equation to get our final answer.
y = –(3/4)x + 2.75
Compare your answer with the correct one above
What is one possible equation for a line parallel to the one passing through the points (4,2) and (15,-4)?
What is one possible equation for a line parallel to the one passing through the points (4,2) and (15,-4)?
(4,2) and (15,-4)
All that we really need to ascertain is the slope of our line. So long as a given answer has this slope, it will not matter what its y-intercept is (given the openness of our question). To find the slope, use the formula: m = rise / run = (y1 - y2) / (x1 - x2):
(2 - (-4)) / (4 - 15) = (2 + 4) / -11 = -6/11
Given this slope, our answer is: y = -6/11x + 57.4
(4,2) and (15,-4)
All that we really need to ascertain is the slope of our line. So long as a given answer has this slope, it will not matter what its y-intercept is (given the openness of our question). To find the slope, use the formula: m = rise / run = (y1 - y2) / (x1 - x2):
(2 - (-4)) / (4 - 15) = (2 + 4) / -11 = -6/11
Given this slope, our answer is: y = -6/11x + 57.4
Compare your answer with the correct one above
What line is parallel to 
at 
?
What line is parallel to
Find the slope of the given line: 
(slope intercept form)
therefore the slope is 
Parallel lines have the same slope, so now we need to find the equation of a line with slope 
and going through point 
by substituting values into the point-slope formula.
So, 
Thus, the new equation is 
Find the slope of the given line:
Parallel lines have the same slope, so now we need to find the equation of a line with slope
So,
Thus, the new equation is
Compare your answer with the correct one above
Lines m and n are parallel
What is the value of angle 
?
Lines m and n are parallel
What is the value of angle
By using the complementary and supplementary rules of geometry (due to lines m and n being parallel), as well as the fact that the sum of all angles within a triangle is 180, we can carry through the operations through stepwise subtraction of 180.
x = 125 → angle directly below also = 125. Since a line is 180 degrees, 180 – 125 = 55. Since right triangle, 90 + 55 = 145 → rightmost angle of triangle 180 – 145 = 35 which is equal to the reflected angle. Use supplementary rule again for 180 – 35 = 145 = y.
Once can also recognize that both a straight line and triangle must sum up to 180 degrees to skip the last step.
By using the complementary and supplementary rules of geometry (due to lines m and n being parallel), as well as the fact that the sum of all angles within a triangle is 180, we can carry through the operations through stepwise subtraction of 180.
x = 125 → angle directly below also = 125. Since a line is 180 degrees, 180 – 125 = 55. Since right triangle, 90 + 55 = 145 → rightmost angle of triangle 180 – 145 = 35 which is equal to the reflected angle. Use supplementary rule again for 180 – 35 = 145 = y.
Once can also recognize that both a straight line and triangle must sum up to 180 degrees to skip the last step.
Compare your answer with the correct one above