Lines - GRE Quantitative Reasoning

Card 0 of 848

Question

what would be the slope of a line perpendicular to

4x+3y = 6

Answer

switch 4x+ 3y = 6 to "y=mx+b" form

3y= -4x + 6

y = -4/3 x + 2

m = -4/3; the perpendicular line will have the negative reciprocal of this line so it would be 3/4

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Question

Which of the following lines is perpendicular to the line ?

Answer

Perpendicular lines will have slopes that are negative reciprocals of one another. Our first step will be to find the slope of the given line by putting the equation into slope-intercept form.

The slope of this line is . The negative reciprocal will be $+\frac{1}{3}$ , which will be the slope of the perpendicular line.

Now we need to find the answer choice with this slope by converting to slope-intercept form.

$y=\frac{4}{12}x-\frac{6}{12}$

$y=\frac{1}{3}x-\frac{1}{2}$

This equation has a slope of $+\frac{1}{3}$ , and must be our answer.

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Question

Which line is perpendicular to the line between the points (22,24) and (31,4)?

Answer

The line will be perpendicular if the slope is the negative reciprocal.

First we need to find the slope of our line between points (22,24) and (31,4). Slope = rise/run = (24 – 4)/(22 – 31) = 20/–9 = –2.22.

The negative reciprocal of this must be a positive fraction, so we can eliminate y = –3_x_ + 5 (because the slope is negative).

The negative reciprocal of –2.22, and therefore the slope of the perpendicular line, will be –1/–2.22 = .45, so we can also eliminate y = x (slope of 1).

Now let's look at the line between points (9, 5) and (48, 19). This slope = (5 – 19)/(9 – 48) = .358, which is incorrect.

The next answer choice is y = .45_x_ + 10. The slope is .45, which is what we're looking for so this is the correct answer.

To double check, the last answer choice is the line between (4, 7) and (7, 4). This slope = (7 – 4) / (4 – 7) = –1, which is also incorrect.

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Question

Which best describes the relationship between the lines $y = \frac{3}{4}x + 8$ and $y = \frac{-4}{3}x + 6$ ?

Answer

We first need to recall the following relationships:

Lines with the same slope and same $\dpi{100} \small y$ -intercept are really the same line.

Lines with the same slope and different $\dpi{100} \small y$ -intercepts are parallel.

Lines with slopes that are negative reciprocals are perpendicular.

Then we identify the slopes of the two lines by comparing the equations to the slope-intercept form $\dpi{100} \small y=mx+b$ , where $\dpi{100} \small m$ is the slope and $\dpi{100} \small b$ is the $\dpi{100} \small y$ -intercept. By inspection we see the lines have slopes of $\dpi{100} \small \frac{3}{4}$ and $\dpi{100} \small \frac{-4}{3}$ . Since these are different, the "parallel" and "same line" choices are eliminated. To test if the slopes are negative reciprocals, we take one of the slopes, change its sign, and flip it upside-down. Starting with $\dpi{100} \small \frac{3}{4}$ and changing the sign gives $\dpi{100} \small \frac{-3}{4}$ , then flipping gives $\dpi{100} \small \frac{-4}{3}$ . This is the same as the slope of the second line, so the two slopes are negative reciprocals and the lines are perpendicular.

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Question

Which of the following equations represents a line that is perpendicular to the line with points and ?

Answer

If lines are perpendicular, then their slopes will be negative reciprocals.

First, we need to find the slope of the given line.

$m=\frac{y_2-y_1}{x_2-x_1}$

$m=\frac{11-(-13)}{7-5}$

$m=\frac{24}{2}$

Because we know that our given line's slope is , the slope of the line perpendicular to it must be $-\frac{1}{12}$ .

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Question

Which of the following lines is perpindicular to

Answer

When determining if a two lines are perpindicular, we are only concerned about their slopes. Consider the basic equation of a line, , where m is the slope of the line. Two lines are perpindicular to each other if one slope is the negative and reciprocal of the other.

The first step of this problem is to get it into the form, , which is . Now we know that the slope, m, is . The reciprocal of that is $\frac{1}{3}$ , and the negative of that is $\frac{-1}{3}$ . Therefore, any line that has a slope of $\frac{-1}{3}$ will be perpindicular to the original line.

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Question

Which of the following lines is perpendicular to the line defined as ?

Answer

To begin, the best thing to do is to put your equation into slope-intercept format. That is, into the format:

For your equation, you need to solve for :

, which is the same as

Then, divide both sides by :

$y=\frac{1}{2}x-\frac{15}{8}$

So, the slope of this line is $\frac{1}{2}$ . The perpendicular of a line is opposite and reciprocal. Therefore, the perpendicular line will have a slope of . Of the options given, only matches this (which you can figure out when you solve for ).

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Question

Which of the following lines is perpendicular to the line passing through the points and ?

Answer

Remember, to be perpendicular, two lines must have opposite and reciprocal slopes. Therefore, you need to begin by solving for the slope of your given line. You do this by finding:

$\frac{rise}{run}$

For two points and , this is:

$\frac{d-b}{c-a}$

For our points, this is:

$\frac{13-4}{4-1}=\frac{9}{3}=3$

The slope of the perpendicular line will be (remember) opposite and reciprocal. Therefore, it will be $-\frac{1}{3}$ . Now, among your equations, the only one that has this slope is:

If you solve this for , you get:

$y=-\frac{1}{3}x + 14$

According to the slope-intercept form (), this means that the slope is $-\frac{1}{3}$ .

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Question

Which of the following lines is perpendicular to the line ?

Answer

Perpendicular lines will have slopes that are negative reciprocals of one another. Our first step will be to find the slope of the given line by putting the equation into slope-intercept form.

The slope of this line is . The negative reciprocal will be $+\frac{1}{3}$ , which will be the slope of the perpendicular line.

Now we need to find the answer choice with this slope by converting to slope-intercept form.

$y=\frac{4}{12}x-\frac{6}{12}$

$y=\frac{1}{3}x-\frac{1}{2}$

This equation has a slope of $+\frac{1}{3}$ , and must be our answer.

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Question

what would be the slope of a line perpendicular to

4x+3y = 6

Answer

switch 4x+ 3y = 6 to "y=mx+b" form

3y= -4x + 6

y = -4/3 x + 2

m = -4/3; the perpendicular line will have the negative reciprocal of this line so it would be 3/4

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Question

Which line is perpendicular to the line between the points (22,24) and (31,4)?

Answer

The line will be perpendicular if the slope is the negative reciprocal.

First we need to find the slope of our line between points (22,24) and (31,4). Slope = rise/run = (24 – 4)/(22 – 31) = 20/–9 = –2.22.

The negative reciprocal of this must be a positive fraction, so we can eliminate y = –3_x_ + 5 (because the slope is negative).

The negative reciprocal of –2.22, and therefore the slope of the perpendicular line, will be –1/–2.22 = .45, so we can also eliminate y = x (slope of 1).

Now let's look at the line between points (9, 5) and (48, 19). This slope = (5 – 19)/(9 – 48) = .358, which is incorrect.

The next answer choice is y = .45_x_ + 10. The slope is .45, which is what we're looking for so this is the correct answer.

To double check, the last answer choice is the line between (4, 7) and (7, 4). This slope = (7 – 4) / (4 – 7) = –1, which is also incorrect.

Compare your answer with the correct one above

Question

Which best describes the relationship between the lines $y = \frac{3}{4}x + 8$ and $y = \frac{-4}{3}x + 6$ ?

Answer

We first need to recall the following relationships:

Lines with the same slope and same $\dpi{100} \small y$ -intercept are really the same line.

Lines with the same slope and different $\dpi{100} \small y$ -intercepts are parallel.

Lines with slopes that are negative reciprocals are perpendicular.

Then we identify the slopes of the two lines by comparing the equations to the slope-intercept form $\dpi{100} \small y=mx+b$ , where $\dpi{100} \small m$ is the slope and $\dpi{100} \small b$ is the $\dpi{100} \small y$ -intercept. By inspection we see the lines have slopes of $\dpi{100} \small \frac{3}{4}$ and $\dpi{100} \small \frac{-4}{3}$ . Since these are different, the "parallel" and "same line" choices are eliminated. To test if the slopes are negative reciprocals, we take one of the slopes, change its sign, and flip it upside-down. Starting with $\dpi{100} \small \frac{3}{4}$ and changing the sign gives $\dpi{100} \small \frac{-3}{4}$ , then flipping gives $\dpi{100} \small \frac{-4}{3}$ . This is the same as the slope of the second line, so the two slopes are negative reciprocals and the lines are perpendicular.

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Question

Which of the following equations represents a line that is perpendicular to the line with points and ?

Answer

If lines are perpendicular, then their slopes will be negative reciprocals.

First, we need to find the slope of the given line.

$m=\frac{y_2-y_1}{x_2-x_1}$

$m=\frac{11-(-13)}{7-5}$

$m=\frac{24}{2}$

Because we know that our given line's slope is , the slope of the line perpendicular to it must be $-\frac{1}{12}$ .

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Question

Which of the following lines is perpindicular to

Answer

When determining if a two lines are perpindicular, we are only concerned about their slopes. Consider the basic equation of a line, , where m is the slope of the line. Two lines are perpindicular to each other if one slope is the negative and reciprocal of the other.

The first step of this problem is to get it into the form, , which is . Now we know that the slope, m, is . The reciprocal of that is $\frac{1}{3}$ , and the negative of that is $\frac{-1}{3}$ . Therefore, any line that has a slope of $\frac{-1}{3}$ will be perpindicular to the original line.

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Question

Which of the following lines is perpendicular to the line defined as ?

Answer

To begin, the best thing to do is to put your equation into slope-intercept format. That is, into the format:

For your equation, you need to solve for :

, which is the same as

Then, divide both sides by :

$y=\frac{1}{2}x-\frac{15}{8}$

So, the slope of this line is $\frac{1}{2}$ . The perpendicular of a line is opposite and reciprocal. Therefore, the perpendicular line will have a slope of . Of the options given, only matches this (which you can figure out when you solve for ).

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Question

Which of the following lines is perpendicular to the line passing through the points and ?

Answer

Remember, to be perpendicular, two lines must have opposite and reciprocal slopes. Therefore, you need to begin by solving for the slope of your given line. You do this by finding:

$\frac{rise}{run}$

For two points and , this is:

$\frac{d-b}{c-a}$

For our points, this is:

$\frac{13-4}{4-1}=\frac{9}{3}=3$

The slope of the perpendicular line will be (remember) opposite and reciprocal. Therefore, it will be $-\frac{1}{3}$ . Now, among your equations, the only one that has this slope is:

If you solve this for , you get:

$y=-\frac{1}{3}x + 14$

According to the slope-intercept form (), this means that the slope is $-\frac{1}{3}$ .

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Question

Given the graph of the line below, find the equation of the line.

Answer

To solve this question, you could use two points such as (1.2,0) and (0,-4) to calculate the slope which is 10/3 and then read the y-intercept off the graph, which is -4.

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Question

What is the equation of the straight line passing through (–2, 5) with an x-intercept of 3?

Answer

First you must figure out what point has an x-intercept of 3. This means the line crosses the x-axis at 3 and has no rise or fall on the y-axis which is equivalent to (3, 0). Now you use the formula (y2 – y1)/(x2 – x1) to determine the slope of the line which is (5 – 0)/(–2 – 3) or –1. Now substitute a point known on the line (such as (–2, 5) or (3, 0)) to determine the y-intercept of the equation y = –x + b. b = 3 so the entire equation is y = –x + 3.

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Question

A line is defined by the following equation:

What is the slope of that line?

Answer

The equation of a line is

y=mx + b where m is the slope

Rearrange the equation to match this:

7x + 28y = 84

28y = -7x + 84

y = -(7/28)x + 84/28

y = -(1/4)x + 3

m = -1/4

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Question

If the coordinates (3, 14) and (_–_5, 15) are on the same line, what is the equation of the line?

Answer

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

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