GRE Math : Coordinate Geometry

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Example Question #1 : How To Find Out If A Point Is On A Line With An Equation

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.

Possible Answers:

Correct answer:

Explanation:

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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Example Question #1 : How To Find Out If A Point Is On A Line With An Equation

For the line

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

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

Possible Answers:

(3, 7)

(9, 5)

(6, –12)

(3, –6)

(6, 5)

Correct answer:

(3, –6)

Explanation:

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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Example Question #91 : Lines

Solve the following system of equations:

–2x + 3y = 10

2x + 5y = 6

Possible Answers:

(–2, 2)

(3, –2)

(2, 2)

(3, 5)

(–2, –2)

Correct answer:

(–2, 2)

Explanation:

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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Example Question #1 : How To Find Out If A Point Is On A Line With An Equation

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

Possible Answers:

$(2,-2)$

$(1,2)$

$(3,4)$

$(2,2)$

$(1,5)$

Correct answer:

$(2,2)$

Explanation:

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

$y=3x-4$

2=3(2)-4

2=6-4

2=2

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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Example Question #98 : Coordinate Geometry

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

Possible Answers:

(1, 0)

(2, 7)

(1, 5)

(-1,2)

(0, 1)

Correct answer:

(1, 5)

Explanation:

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.

y=3x+2

5=3(1)+2

5=3+2

5=5

Because this equality is true, we can conclude that the point (1,5) lies on this line. None of the other given answer options will result in a true equality.

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

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

Possible Answers:

y = –x + 3

y = –5x + 3

y = –x – 3

y = –5x – 3

Correct answer:

y = –x + 3

Explanation:

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.

Gre_quant_179_01

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Example Question #2 : How To Find The Equation Of A Line

What is the equation of the line passing through (–1,5) and the upper-right corner of a square with a center at the origin and a perimeter of 22?

Possible Answers:

y = (–3/5)x + 22/5

y = (3/5)x + 22/5

y = –x + 5

y = (–3/5)x + 28/5

y = (–1/5)x + 2.75

Correct answer:

y = (–3/5)x + 22/5

Explanation:

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

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

What is the equation of a line passing through the points (-2,17) and (5,-11) ?

Possible Answers:

y=-4x+9

$y=\frac{14}{3}x+17$

y=4x+17

y=-27x-15

$y=\frac{12}{5}x-20$

Correct answer:

y=-4x+9

Explanation:

Based on the information provided, you can find the slope of this line easily. From that, you can use the point-slope form of the equation of a line to compute the line's full equation. The slope is merely:

$\frac{rise}{run}=\frac{17-(-11)}{-2-5}=\frac{17+11}{-7}=\frac{-28}{7}=-4$

Now, for a point (a,b) and a slope , the point-slope form of a line is:

y-b=m(x-a)

Let's use (-2,17) for our point (a,b)

This gives us:

y-17=-4(x+2)

Now, distribute and solve for :

y-17=-4x-8

y=-4x+9

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Example Question #4 : How To Find The Equation Of A Line

What is the equation of a line passing through the two points (41,11) and (4,-9) ?

Possible Answers:

y=14x-18

$y=\frac{17}{14}x-\frac{148}{25}$

$y=\frac{20}{37}x-\frac{413}{37}$

$y=\frac{20}{27}x-\frac{14}{15}$

$y=\frac{7}{2}x-\frac{85}{3}$

Correct answer:

$y=\frac{20}{37}x-\frac{413}{37}$

Explanation:

$\frac{rise}{run}=\frac{11-(-9)}{41-4}=\frac{11+9}{37}=\frac{20}{37}$

Now, for a point (a,b) and a slope , the point-slope form of a line is:

y-b=m(x-a)

Let's use (4,-9) for our point (a,b)

This gives us:

$y+9=\frac{20}{37}(x-4)$

Now, distribute and solve for :

$y+9=\frac{20}{37}x-\frac{80}{37}$

$y=\frac{20}{37}x-\frac{80}{37}-9$

$y=\frac{20}{37}x-\frac{80}{37}-\frac{333}{37}$

$y=\frac{20}{37}x-\frac{413}{37}$

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Example Question #5 : How To Find The Equation Of A Line

What is the equation of a line passing through (4,-12) with a -intercept of ?

Possible Answers:

$y=\frac{-15}{4}x+12$

y=-22x+15

$y=\frac{-21}{4}x+9$

$y=\frac{27}{5}x+9$

y=-5x-9

Correct answer:

$y=\frac{-21}{4}x+9$

Explanation:

Based on the information that you have been provided, you can quickly find the slope of your line. Since the y-intercept is , you know that the line contains the point (0,9) . Therefore, the slope of the line is found:

$\frac{-12-9}{4-0}=\frac{-21}{4}$

Based on this information, you can use the standard slope-intercept form to find your equation:

y=mx+b , where $m=\frac{-21}{4}$ and b=9

$y=\frac{-21}{4}x+9$

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GRE Math : Coordinate Geometry

Study concepts, example questions & explanations for GRE Math

All GRE Math Resources

Example Questions

Example Question #1 : How To Find Out If A Point Is On A Line With An Equation

Example Question #1 : How To Find Out If A Point Is On A Line With An Equation

Example Question #91 : Lines

Example Question #1 : How To Find Out If A Point Is On A Line With An Equation

Example Question #98 : Coordinate Geometry

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

Example Question #2 : How To Find The Equation Of A Line

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

Example Question #4 : How To Find The Equation Of A Line

Example Question #5 : How To Find The Equation Of A Line

All GRE Math Resources

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