GRE Math : Coordinate Geometry

Study concepts, example questions & explanations for GRE Math

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Example Questions

Example Question #1401 : Gre Quantitative Reasoning

A certain line has points at \(\displaystyle (3,-5)\) and \(\displaystyle (-6,1)\). Which of the following lines is parallel to the so-called "certain" line?

Possible Answers:

\(\displaystyle y=-\frac{2}{3}x + 6\)

\(\displaystyle y=-\frac{3}{2}x + \frac{1}{2}\)

\(\displaystyle y=\frac{3}{2}x - 2\)

\(\displaystyle y = -2x - 3\)

Correct answer:

\(\displaystyle y=-\frac{2}{3}x + 6\)

Explanation:

To begin, we must first solve the for the slope of the original line. Use the formula for slope to do this:

\(\displaystyle slope=\frac{rise}{run}=\frac{\triangle y}{\triangle x}=\frac{y_2-y_1}{x_2-x_1}\)

Use the two points we were given:

\(\displaystyle slope=\frac{y_2-y_1}{x_2-x_1}\)

\(\displaystyle =\frac{1-(-5)}{(-6)-3}\)

\(\displaystyle =\frac{1+5}{-6-3}\)

\(\displaystyle =\frac{6}{-9}\)

Reduce the fraction:

\(\displaystyle =\frac{2}{-3}\)

\(\displaystyle =-\frac{2}{3}\)

We are looking for any answer choice with a slope of \(\displaystyle -\frac{2}{3}\). The answer is \(\displaystyle y=-\frac{2}{3}x + 6\).

Example Question #1 : How To Find Out If Lines Are Parallel

There are two lines:

2x – 4y = 33

2x + 4y = 33

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

Possible Answers:

Perpendicular

Parallel

None of the other answers

Non-perpendicular intersecting

The same

Correct answer:

Non-perpendicular intersecting

Explanation:

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.

Example Question #1 : How To Find Out If Lines Are Parallel

\(\displaystyle A: 2x+7y=14\)

\(\displaystyle B: 14x=7-2y\)

\(\displaystyle C:91-14y=4x\)

\(\displaystyle D: 10.5y=814-3x\)

Which of the above-listed lines are parallel?

Possible Answers:

\(\displaystyle A\) and \(\displaystyle C\)

\(\displaystyle B\) and \(\displaystyle D\)

All four lines

\(\displaystyle A\)\(\displaystyle C\), and \(\displaystyle D\)

None of them

Correct answer:

\(\displaystyle A\)\(\displaystyle C\), and \(\displaystyle D\)

Explanation:

There are several ways to solve this problem.  You could solve all of the equations for \(\displaystyle y\).  This would give you equations in the form \(\displaystyle y=mx+b\).  All of the lines with the same \(\displaystyle m\) value would be parallel.  Otherwise, you could figure out the ratio of \(\displaystyle y\) to \(\displaystyle x\) 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 \(\displaystyle y\):

\(\displaystyle A: 2x+7y=14\)

\(\displaystyle 7y=14-2x\)

\(\displaystyle y=2-\frac{2}{7}x\)

 

\(\displaystyle B: 14x=7-2y\)

\(\displaystyle 2y=7-14x\)

\(\displaystyle y=\frac{7}{2}-7x\)

 

\(\displaystyle C:91-14y=4x\)

\(\displaystyle -14y=4x-91\)

\(\displaystyle y=\frac{4x}{-14}-\frac{91}{-14}\)

\(\displaystyle y=\frac{13}{2}-\frac{2x}{7}\)

 

\(\displaystyle D: 10.5y=814-3x\)

\(\displaystyle y=\frac{814}{10.5}-\frac{3}{10.5}x\)

Here, you need to be a bit more manipulative with your equation.  Multiply the numerator and denominator of the \(\displaystyle x\) value by \(\displaystyle 2\):

\(\displaystyle y=\frac{814}{10.5}-\frac{6}{21}x\)

\(\displaystyle y=\frac{814}{10.5}-\frac{2}{7}x\)

 

Therefore, \(\displaystyle A\)\(\displaystyle C\), and \(\displaystyle D\) all have slopes of \(\displaystyle -\frac{2}{7}\)

Example Question #21 : Lines

Which of the following is parallel to the line passing through \(\displaystyle (4,3)\) and \(\displaystyle (6,3)\)?

Possible Answers:

\(\displaystyle x=6\)

\(\displaystyle 5y-15x=4\)

\(\displaystyle 4x-2y=8\)

\(\displaystyle 3y=3x+6\)

\(\displaystyle y=4\)

Correct answer:

\(\displaystyle y=4\)

Explanation:

Now, notice that the slope of the line that you have been given is \(\displaystyle 0\). You know this because slope is merely:

\(\displaystyle \frac{rise}{run}\)

However, for your points, there is no rise at all. You do not even need to compute the value. You know it will be \(\displaystyle 0\). All lines with slope \(\displaystyle 0\) are of the form \(\displaystyle y=c\), where \(\displaystyle c\) is the value that \(\displaystyle y\) has for all \(\displaystyle x\) points. Based on our data, this is \(\displaystyle y=3\), for \(\displaystyle y\) is always \(\displaystyle 3\)—no matter what is the value for \(\displaystyle x\). So, the parallel answer choice is \(\displaystyle y=4\), as both have slopes of \(\displaystyle 0\).

Example Question #2 : How To Find Out If Lines Are Parallel

Which of the following is parallel to \(\displaystyle 3y+6x=12\)?

Possible Answers:

The line between the points \(\displaystyle (-1,7)\) and \(\displaystyle (3,8)\)

The line between the points \(\displaystyle (5,3)\) and \(\displaystyle (7,7)\)

The line between the points \(\displaystyle (1,1)\) and \(\displaystyle (3,2)\)

The line between the points \(\displaystyle (1,4)\) and \(\displaystyle (5,1)\)

The line between the points \(\displaystyle (4,7)\) and \(\displaystyle (6,3)\)

Correct answer:

The line between the points \(\displaystyle (4,7)\) and \(\displaystyle (6,3)\)

Explanation:

\(\displaystyle 3y+6x=12\)

To begin, solve your equation for \(\displaystyle y\). This will put it into slope-intercept form, which will easily make the slope apparent. (Remember, slope-intercept form is \(\displaystyle y=mx+b\), where \(\displaystyle m\) is the slope.)

\(\displaystyle 3y=-6x+12\)

Divide both sides by \(\displaystyle 3\) and you get:

\(\displaystyle y=-2x+4\)

Therefore, the slope is \(\displaystyle -2\). Now, you need to test your points to see which set of points has a slope of \(\displaystyle -2\). Remember, for two points \(\displaystyle (a,b)\) and \(\displaystyle (c,d)\), you find the slope by using the equation:

\(\displaystyle \frac{rise}{run}=\frac{d-b}{c-a}\)

For our question, the pair \(\displaystyle (6,3)\) and \(\displaystyle (4,7)\) gives us a slope of \(\displaystyle -2\):

\(\displaystyle \frac{7-3}{4-6}=\frac{4}{-2}=-2\)

Example Question #223 : New Sat

Which of the following lines is parallel to:

\(\displaystyle 4y-12x=2\)

 

Possible Answers:

\(\displaystyle y=\frac{1}{3}x+2\)

\(\displaystyle x-\frac{1}{3}y=7\)

\(\displaystyle x-3y=9\)

\(\displaystyle x+4y=10\)

Correct answer:

\(\displaystyle x-\frac{1}{3}y=7\)

Explanation:

First write the equation in slope intercept form. Add \(\displaystyle 12x\) to both sides to get \(\displaystyle 4y = 12x + 2\). Now divide both sides by \(\displaystyle 4\) to get \(\displaystyle y = 3x + 0.5\). The slope of this line is \(\displaystyle 3\), so any line that also has a slope of \(\displaystyle 3\) would be parallel to it. The correct answer is  \(\displaystyle x - \frac{1}{3}y = 7\).

Example Question #1 : How To Find Out If Lines Are Parallel

Which pair of linear equations represent parallel lines?

Possible Answers:

y=2x+4\(\displaystyle y=2x+4\)

y=x+4\(\displaystyle y=x+4\)

y=x-5\(\displaystyle y=x-5\)

y=3x+5\(\displaystyle y=3x+5\)

y=2x-4\(\displaystyle y=2x-4\)

y=2x+5\(\displaystyle y=2x+5\)

y=x+2\(\displaystyle y=x+2\)

y=-x+2\(\displaystyle y=-x+2\)

y=-x+4\(\displaystyle y=-x+4\)

y=x+6\(\displaystyle y=x+6\)

Correct answer:

y=2x-4\(\displaystyle y=2x-4\)

y=2x+5\(\displaystyle y=2x+5\)

Explanation:

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\(\displaystyle m\)" spot in the linear equation (y=mx+b)\(\displaystyle (y=mx+b)\)

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

Example Question #25 : Coordinate Geometry

Which of the following equations represents a line that is parallel to the line represented by the equation \(\displaystyle 10x-4y=26\)?

Possible Answers:

\(\displaystyle y=\frac{5}{2}x+1\)

\(\displaystyle y=-\frac{5}{2}x+1\)

\(\displaystyle y=-\frac{2}{5}x+3\)

\(\displaystyle y=\frac{2}{5}x+2\)

\(\displaystyle y=5x-1\)

Correct answer:

\(\displaystyle y=\frac{5}{2}x+1\)

Explanation:

Lines are parallel when their slopes are the same.

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

\(\displaystyle -4y=-10+26\)

\(\displaystyle y=\frac{10}{4}x-\frac{26}{4}\)

\(\displaystyle y=\frac{5}{2}x-\frac{13}{2}\)

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

Example Question #1 : Other Lines

Refer to the following graph:

Gre1

What is the slope of the line shown?

Possible Answers:

–3

–1

1/3

3

–1/3

Correct answer:

–3

Explanation:

One can use either the slope formula m = (y2 – y1)/(x2 – x1) or the standard line equation, y = mx + b to solve for the slope, m. By calculation or observation, one can determine that the slope is –3.

Example Question #1 : How To Find The Slope Of A Line

What is the slope of the equation 4x + 3y = 7?

Possible Answers:

4/3

–4/3

3/4

–3/4

–7/3

Correct answer:

–4/3

Explanation:

We should put this equation in the form of y = mx + b, where m is the slope.

We start with 4x + 3y = 7.

Isolate the y term: 3y = 7 – 4x

Divide by 3: y = 7/3 – 4/3 * x

Rearrange terms: y = –4/3 * x + 7/3, so the slope is –4/3.

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