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SAT Math : Coordinate Geometry

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Example Questions

Example Question #508 : Geometry

Solve each problem and decide which is the best of the choices given.

Find the slope of the line for the given equation.
12x+3y=-6

Possible Answers:

$\frac{1}{4}$

$-\frac{1}{2}$

Correct answer:

Explanation:

For this problem, you have to solve for . We want to get the equation in slope-intercept form,

y=mx+b where represents the slope of the line.

12x+3y=-6

First subtract 12x from each side to get

3y=-12x-6 .

Then divide both sides by to get

y=-4x-2 .

The slope is the number in front of , so the slope is .

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Example Question #504 : Geometry

Point is at (-2,-5) and point is at (3,1) . What is the slope of the line that connects the two points?

Possible Answers:

$\frac{6}{5}$

$-\frac{2}{3}$

$\frac{3}{2}$

$\frac{1}{5}$

$-\frac{6}{5}$

Correct answer:

$\frac{6}{5}$

Explanation:

The purpose of this question is to understand how the slope of a line is calculated.

The slope is the rise over the run, meaning the change in the y values over the change in the x values

$\frac{y_2-y_1}{x_2-x_1}$ .

So, the difference in y values divided by the difference in x values yields

$\frac{1-(-5)}{3-(-2)}=\frac{6}{5}$ .

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Example Question #510 : Geometry

The following two points are located on the same line. What is the slope of the line?

$(2\pi, 5\pi), (11\pi, 6\pi)$

Possible Answers:

$9\pi$

$\frac{1}{9}$

$\frac{1}{9}\pi$

Correct answer:

$\frac{1}{9}$

Explanation:

The slope of a line with two points (x_1, y_1) and (x_2, y_2) is given by the following equation:

$m = \frac{y_2-y_1}{x_2-x_1}$

Let $(x_1, y_1) = (2\pi, 5\pi)$ and $(x_2, y_2) = (11\pi, 6\pi)$ . Substituting these values into the equation gives us:

$m = \frac{6\pi - 5\pi}{11\pi - 2\pi} = \frac{\pi}{9\pi} = \frac{1}{9}$

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Example Question #15 : How To Find Slope Of A Line

Axes

Figure NOT drawn to scale

On the coordinate axes shown above, the shaded triangle has area 16.

Give the slope of the line that includes the hypotenuse of the triangle.

Possible Answers:

$-\frac{1}{2}$

None of the other choices gives the correct response.

$-\frac{1}{4}$

Correct answer:

Explanation:

The length of the horizontal leg of the triangle is the distance from the origin (0,0) to (4,0) , which is 4.

The area of a right triangle is half the product of the lengths of its legs and , so, setting a = 4 and A = 16 and solving for :

$\frac{1}{2} ab = A$

$\frac{1}{2} \cdot 4 \cdot b = 16$

2b = 16

$\frac{2b }{2}= \frac{16}{2}$

b = 8

Since this is the vertical distance from the origin, this is also the absolute value of the -coordinate of the -intercept of the line; also, this point is along the positive -axis. The line has -intercept (0,8) .

The slope of a line, given the intercepts (a,0) , (0, b) , is

$m = - \frac{b}{a}$ ,

Substitute a = 4 and b = 8 :

$m = - \frac{8}{4} = -2$ .

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Example Question #121 : Coordinate Geometry

A line passes through the origin, and the points (1, k) and (k,4). What is a possible value for k?

Possible Answers:

k=4

k=1

k=2

k=-4

k=0

Correct answer:

k=2

Explanation:

To find k, first identify what is known.

The origin means the point (0,0).

Therefore the line passes through the points:

(0,0), (1,k),(k,4)

Since a line by definition has the same slope between the points, calculation the slope between the origin and each of the points.

$\text{slope}=\frac{y_2-y_1}{x_2-x_1}$

Let

$\\(x_1,y_1)=(0,0) \\(x_2,y_2)=(1,k)$

therefore the slope is,

$\text{slope}=\frac{k-0}{1-0}=k$

Now let,

$\\(x_1,y_1)=(0,0) \\(x_2,y_2)=(k,4)$

thus the slope is,

$\text{slope}=\frac{4-0}{k-0}=\frac{4}{k}$

From here set the slopes equal to each other and solve for k.

$k=\frac{4}{k}$

Multiply by k on both sides.

$\\k\cdot k=\frac{4}{k}\cdot k \\\\k^2=4 \\\\\sqrt{k^2}=\sqrt{4} \\\\k=\pm2$

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Example Question #512 : Sat Mathematics

Axes

Figure NOT drawn to scale

On the coordinate axes shown above, the shaded triangle has area 60.

Give the slope of the line that includes the hypotenuse of the triangle.

Possible Answers:

$- \frac{12}{5}$

$- \frac{5}{12}$

$- \frac{6}{5}$

$- \frac{5}{6}$

$\textup{None of the other choices gives the correct response}$

Correct answer:

$- \frac{6}{5}$

Explanation:

The length of the vertical leg of the triangle is the distance from the origin (0,0) to (0, 12) , which is 12.

The area of a right triangle is half the product of the lengths of its legs and , so, setting b= 12 and A = 60 and solving for :

$\frac{1}{2} ab = A$

$\frac{1}{2} \cdot a \cdot 12 = 60$

6a = 60

$\frac{6a }{6}= \frac{60}{6}$

a= 10

Since this is the positive horizontal distance from the origin, this is also the -coordinate of the -intercept of the line - that is, the line has -intercept (10,0) .

The slope of a line, given the intercepts (a,0) , (0, b) , is

$m = - \frac{b}{a}$ ,

Substitute a= 10 and :

$m = - \frac{12}{10} = - \frac{6}{5}$

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Example Question #2 : Coordinate Geometry

Solve the equation for x and y.

–x – 4y = 245

5x + 2y = 150

Possible Answers:

x = 234/5

y = 1245/15

x = –1375/9

y = 545/18

x = 3

y = 7

x = 545/9

y = –1375/18

Correct answer:

x = 545/9

y = –1375/18

Explanation:

While solving the problem requires the same method as the ones above, this is one is more complicated because of the more complex given equations. Start of by deriving a substitute for one of the unknowns. From the second equation we can derive y=75-(5x/2). Since 2y = 150 -5x, we divide both sides by two and find our substitution for y. Then we enter this into the first equation. We now have –x-4(75-(5x/2))=245. Distribute the 4. So we get –x – 300 + 10x = 245. So 9x =545, and x=545/9. Use this value for x and solve for y. The graph below illustrates the solution.

Sat_math_165_04

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Example Question #11 : Psat Mathematics

Solve the equation for x and y.

y + 5x = 40

x – y = –10

Possible Answers:

x = 7

y = 12

x = 15

y = 5

x = 5

y = 15

x = 12

y = 7

Correct answer:

x = 5

y = 15

Explanation:

This one is a basic problem with two unknowns in two equations. Derive y=x+10 from the second equation and replace the y in first equation to solve the problem. So, x+10+5x=40 and x = 5. X-y= -10 so y=15. The graph below illustrates the solution.

Sat_math_165_05

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Example Question #33 : Other Lines

Solve the equation for x and y.

xy=30

x – y = –1

Possible Answers:

x = 2, 3

y = 4, 5

x = –7, 3

y = 7, –3

x = 4, 5

y = –4, –5

x = 5, –6

y = 6, –5

Correct answer:

x = 5, –6

y = 6, –5

Explanation:

Again the same process is required. This problem however involved multiplying x by y so is a bit different. We end up with two possible solutions. Derive y=x+1 and solve in the same manner as the ones above. The graph below illustrates the solution.

Sat_math_165_07

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

Solve the equation for x and y.

x/y = 30

x + y = 5

Possible Answers:

x = 5/150

y = 150/31

x = 150/31

y = 5/31

x = 7

y = 14

x = 2, 6

y = 3, 7

Correct answer:

x = 150/31

y = 5/31

Explanation:

Similar problem to the one before, with x being divided by y instead of multiplied. Solve in the same manner but keep in mind the way that x/y is graphed. We end up solving for one solution. The graph below illustrates the solution,

Sat_math_165_08

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SAT Math : Coordinate Geometry

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #508 : Geometry

Example Question #504 : Geometry

Example Question #510 : Geometry

Example Question #15 : How To Find Slope Of A Line

Example Question #121 : Coordinate Geometry

Example Question #512 : Sat Mathematics

Example Question #2 : Coordinate Geometry

Example Question #11 : Psat Mathematics

Example Question #33 : Other Lines

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

All SAT Math Resources

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