Coordinate Geometry - GRE Quantitative Reasoning

Card 0 of 20

Question

For the line

$y=\frac{1}{3}x-7$

Which one of these coordinates can be found on the line?

Answer

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

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Question

Consider the lines described by the following two equations:

4y = 3x2

3y = 4x2

Find the vertical distance between the two lines at the points where x = 6.

Answer

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.

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Question

Solve the following system of equations:

–2x + 3y = 10

2x + 5y = 6

Answer

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)

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Question

Find the point where the line y = .25(x – 20) + 12 crosses the x-axis.

Answer

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).

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Question

Determine the greater quantity:

or

Answer

$\dpi{100} \small BD+AC$ is the length of the line, except that $\dpi{100} \small BC$ is double counted. By subtracting $\dpi{100} \small BC$ , we get the length of the line, or $\dpi{100} \small AD$ .

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Question

Which of the following sets of coordinates are on the line $y=3x-4$ ?

Answer

$(2,2)$ when plugged in for $y$ and $x$ make the linear equation true, therefore those coordinates fall on that line.

$y=3x-4$

Because this equation is true, the point must lie on the line. The other given answer choices do not result in true equalities.

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Question

Which of the following points can be found on the line $\small y=3x+2$ ?

Answer

We are looking for an ordered pair that makes the given equation true. To solve, plug in the various answer choices to find the true equality.

Because this equality is true, we can conclude that the point lies on this line. None of the other given answer options will result in a true equality.

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Question

On a coordinate plane, two lines are represented by the equations and . These two lines intersect at point . What are the coordinates of point ?

Answer

You can solve for the within these two equations by eliminating the . By doing this, you get .

Solve for to get and plug back into either equation to get the value of as 1.

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Question

If the two lines represented by and intersect at point , what are the coordinates of point ?

Answer

Solve for by setting the two equations equal to one another:

Plugging back into either equation gives .

These are the coordinates for the intersection of the two lines.

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Question

Which of the following set of points is on the line formed by the equation ?

Answer

The easy way to solve this question is to take each set of points and put it into the equation. When we do this, we find the only time the equation balances is when we use the points .

For practice, try graphing the line to see which of the points fall on it.

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Question

Which of the following lines is parallel to:

Answer

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 $x - \frac{1}{3}y = 7$ .

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Question

There are two lines:

2x – 4y = 33

2x + 4y = 33

Are these lines perpendicular, parallel, non-perpendicular intersecting, or the same lines?

Answer

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.

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Question

Which pair of linear equations represent parallel lines?

Answer

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 " $m$ " spot in the linear equation $(y=mx+b)$ ,

We are looking for an answer choice in which both equations have the same $m$ value. Both lines in the correct answer have a slope of 2, therefore they are parallel.

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Question

Which of the following equations represents a line that is parallel to the line represented by the equation ?

Answer

Lines are parallel when their slopes are the same.

First, we need to place the given equation in the slope-intercept form.

$y=\frac{10}{4}x-\frac{26}{4}$

$y=\frac{5}{2}x-\frac{13}{2}$

Because the given line has the slope of $\frac{5}{2}$ , the line parallel to it must also have the same slope.

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Question

Which of the above-listed lines are parallel?

Answer

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 :

$y=2-\frac{2}{7}x$

$y=\frac{7}{2}-7x$

$y=\frac{4x}{-14}-\frac{91}{-14}$

$y=\frac{13}{2}-\frac{2x}{7}$

$y=\frac{814}{10.5}-\frac{3}{10.5}x$

Here, you need to be a bit more manipulative with your equation. Multiply the numerator and denominator of the value by :

$y=\frac{814}{10.5}-\frac{6}{21}x$

$y=\frac{814}{10.5}-\frac{2}{7}x$

Therefore, , , and all have slopes of $-\frac{2}{7}$

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Question

Which of the following is parallel to the line passing through and ?

Answer

Now, notice that the slope of the line that you have been given is . You know this because slope is merely:

$\frac{rise}{run}$

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 .

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Question

Which of the following is parallel to ?

Answer

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:

$\frac{rise}{run}=\frac{d-b}{c-a}$

For our question, the pair and gives us a slope of :

$\frac{7-3}{4-6}=\frac{4}{-2}=-2$

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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.

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