Coordinate Geometry - GRE Quantitative Reasoning

To find the slope, put the equation in slope-intercept form . In this case we have , which indicates that the slope is .

distance2 = (_x_2 – _x_1)2 + (_y_2 – _y_1)2 + (_z_2 – _z_1)2

= (4 – 2)2 + (6 – 3)2 + (5 – 4)2

= 22 + 32 + 12

= 14

distance = √14

The slope is equal to the difference between the y-coordinates divided by the difference between the x-coordinates.

Use the give points in this formula to calculate 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 .

A question like this is actually rather easy. All you need to do is rewrite the equation in slope intercept form, that is:

Therefore, begin to simplify:

Becomes...

Then...

Finally, divide both sides by :

The coefficient for the term is your slope:

There are two key facts to register about this drawing. First, the line clearly has a negative slope, given that it runs "downhill" when you look at it from left to right. Secondly, it has a positive y-intercept. Therefore, you know that the coefficient for the term must be negative, and the numerical coefficient for the y-intercept must be positive. This only occurs in the equation . Therefore, this is the only viable option.

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:

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).

First rearrange the equation so that it is in slope-intercept form, resulting in y=2/3 x + 14/9. The slope of this line is 2/3, so the slope of the line perpendicular will have the opposite reciprocal as a slope, which is -3/2.

Perpendicular lines have slopes that are the opposite of the reciprocal of each other. In this case, the slope of the first line is -2. The reciprocal of -2 is -1/2, so the opposite of the reciprocal is therefore 1/2.

distance2 = (_x_2 – _x_1)2 + (_y_2 – _y_1)2

Looking at the two order pairs given, _x_1 = 1, _y_1 = 1, _x_2 = 7, _y_2 = 9.

distance2 = (7 – 1)2 + (9 – 1)2 = 62 + 82 = 100

distance = 10

Recall that the midpoint formula is like finding the average of the and values for two points. For two points and , it is:

For our points, we are looking for . We know:

We can solve for each of these coordinates separately:

X-Coordinate

Y-Coordinate:

Therefore, our point is

Since the equation is defined as it is, you know the y-intercept is . This is the point . To find the slope of the line, you merely need to use the two points that you have and find the equation:

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

To find if something is perpendicular, you need to first know the slope of your given line. Based on your points, this is easy. Recall that slope is merely:

This is:

Since a perpendicular line has a slope that is both opposite in sign and reciprocal, you need to choose a line with a slope of . The only possible option is, therefore,

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