Lines - GRE Quantitative Reasoning

To test the coordinates, plug the x-coordinate into the line equation and solve for y.

y = 1/3x -7

Test (3,-6)

y = 1/3(3) – 7 = 1 – 7 = -6 YES!

Test (3,7)

y = 1/3(3) – 7 = 1 – 7 = -6 NO

Test (6,-12)

y = 1/3(6) – 7 = 2 – 7 = -5 NO

Test (6,5)

y = 1/3(6) – 7 = 2 – 7 = -5 NO

Test (9,5)

y = 1/3(9) – 7 = 3 – 7 = -4 NO

Solve equation for y. y=mx+b, where m is the slope

To find the slope of a line, you should convert an equation to the slope-intercept form. In this case, the equation would be , which means the slope is .

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

(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

To test the coordinates, plug the x-coordinate into the line equation and solve for y.

y = 1/3x -7

Test (3,-6)

y = 1/3(3) – 7 = 1 – 7 = -6 YES!

Test (3,7)

y = 1/3(3) – 7 = 1 – 7 = -6 NO

Test (6,-12)

y = 1/3(6) – 7 = 2 – 7 = -5 NO

Test (6,5)

y = 1/3(6) – 7 = 2 – 7 = -5 NO

Test (9,5)

y = 1/3(9) – 7 = 3 – 7 = -4 NO

Solve equation for y. y=mx+b, where m is the slope

To find the slope of a line, you should convert an equation to the slope-intercept form. In this case, the equation would be , which means the slope is .

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

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.

We first need to recall the following relationships:

Lines with the same slope and same -intercept are really the same line.

Lines with the same slope and different -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 , where is the slope and is the -intercept. By inspection we see the lines have slopes of and . 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 and changing the sign gives , then flipping gives . This is the same as the slope of the second line, so the two slopes are negative reciprocals and the lines are perpendicular.

Recall that to find the midpoint of two points and , you use the equation:

.

(It is just like finding the average of the two points, really.)

So, for our equation, we know the following:

You merely need to solve each coordinate for its respective value.

Then, for the y-coordinate:

Therefore, our other point is:

Recall that to find the midpoint of two points and , you use the equation:

.

(It is just like finding the average of the two points, really.)

So, for our equation, we know the following:

You merely need to solve each coordinate for its respective value.

Then, for the y-coordinate:

Therefore, our other point is:

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

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

Because we know that our given line's slope is , the slope of the line perpendicular to it must be .

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 , and the negative of that is . Therefore, any line that has a slope of will be perpindicular to the original line.

Since the vertical coordinates of each point are given by y, solve each equation for y and plug in 6 for x, as follows:

Taking the difference of the resulting y -values give the vertical distance between the points (6,27) and (6,48), which is 21.

Since we have –2x and +2x in the equations, it makes sense to add the equations together to give 8y = 16 yielding y = 2. Then we substitute y = 2 into one of the original equations to get x = –2. So the solution to the system of equations is (–2, 2)

When the line crosses the x-axis, the y-coordinate is 0. Substitute 0 into the equation for y and solve for x.

.25(x – 20) + 12 = 0

.25_x_ – 5 = –12

.25_x_ = –7

x = –28

The answer is the point (–28,0).

is the length of the line, except that is double counted. By subtracting , we get the length of the line, or .