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SAT Math : Coordinate Geometry

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Example Questions

Example Question #1 : Lines

Which of the following is the equation of a line that is parallel to the line 4x – y = 22 and passes through the origin?

Possible Answers:

(1/4)x + y = 0

4x – y = 0

4x + 8y = 0

y – 4x = 22

4x = 8y

Correct answer:

4x – y = 0

Explanation:

We start by rearranging the equation into the form y = mx + b (where m is the slope and b is the y intercept); y = 4x – 22
Now we know the slope is 4 and so the equation we are looking for must have the m = 4 because the lines are parallel. We are also told that the equation must pass through the origin; this means that b = 0.

In 4x – y = 0 we can rearrange to get y = 4x. This fulfills both requirements.

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Example Question #1 : Lines

What line is parallel to 2x + 5y = 6 through (5, 3)?

Possible Answers:

y = 5/3x – 5

y = –2/5x + 5

y = 5/2x + 3

y = 3/5x – 2

y = –2/3x + 3

Correct answer:

y = –2/5x + 5

Explanation:

The given equation is in standard form and needs to be converted to slope-intercept form which gives y = –2/5x + 6/5. The parallel line will have a slope of –2/5 (the same slope as the old line). The slope and the given point are substituted back into the slope-intercept form to yield y = –2/5x +5.

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Example Question #2 : Lines

What line is parallel to y=2x-1 through (1,-3) ?

Possible Answers:

x-2y=6

2x-y=7

3x+2y=-4

-2x+y=-5

x+3y=11

Correct answer:

-2x+y=-5

Explanation:

The slope of the given line is m=2 and a parallel line would have the same slope, so we need to find a line through (1,-3) with a slope of 2 by using the slope-intercept form of the equation for a line. The resulting line is y=2x-5 which needs to be converted to the standard form to get -2x+y=-5 .

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Example Question #1 : Parallel Lines

There is a line defined by the equation below:

3x+4y=12

There is a second line that passes through the point (1,2) and is parallel to the line given above. What is the equation of this second line?

Possible Answers:

$y=\frac{3}{4}x+2.75$

$y=-\frac{3}{4}x+1.25$

$y=-\frac{3}{4}x+2.75$

$y=\frac{3}{4}x+1.25$

$y=\frac{3}{4}x+2.625$

Correct answer:

$y=-\frac{3}{4}x+2.75$

Explanation:

Parallel lines have the same slope. Solve for the slope in the first line by converting the equation to slope-intercept form.

3x + 4y = 12

4y = –3x + 12

y = –(3/4)x + 3

slope = –3/4

We know that the second line will also have a slope of –3/4, and we are given the point (1,2). We can set up an equation in slope-intercept form and use these values to solve for the y-intercept.

y = mx + b

2 = –3/4(1) + b

2 = –3/4 + b

b = 2 + 3/4 = 2.75

Plug the y-intercept back into the equation to get our final answer.

y = –(3/4)x + 2.75

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Example Question #2 : Parallel Lines

What is the equation of a line that is parallel to 3x + 5y = 8 and passes through (10, 4) ?

Possible Answers:

$y=\frac{3}{4}x-2$

$y=\frac{5}{3}x+5$

$y=-\frac{3}{5}x+10$

$y=\frac{5}{6}x-10$

$y=\frac{3}{8}x-5$

Correct answer:

$y=-\frac{3}{5}x+10$

Explanation:

To solve, we will need to find the slope of the line. We know that it is parallel to the line given by the equation, meaning that the two lines will have equal slopes. Find the slope of the given line by converting the equation to slope-intercept form.

3x + 5y = 8

5y=-3x+8

$y=-\frac{3}{5}x+\frac{8}{5}$

The slope of the line will be $-\frac{3}{5}$ . In slope intercept-form, we know that the line will be $y=-\frac{3}{5}x+b$ . Now we can use the given point to find the y-intercept.

$y=-\frac{3}{5}x+b$

$4=-\frac{3}{5}(10)+b$

4=-6+b

10=b

The final equation for the line will be $y=-\frac{3}{5}x+10$ .

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Example Question #1 : How To Find The Equation Of Parallel Lines

What line is parallel to 3x + 2y = 6 and passes through the point (2,-1) ?

Possible Answers:

$y = \frac{-1}{3}x - 5$

y = 3x - 15

$y = \frac{1}{3}x + 6$

$y = \frac{-3}{2}x + 2$

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

Correct answer:

$y = \frac{-3}{2}x + 2$

Explanation:

Start by converting the original equation to slop-intercept form.

3x + 2y = 6

2y=-3x+6

$y=-\frac{3}{2}+3$

The slope of this line is $-\frac{3}{2}$ . A parallel line will have the same slope. Now that we know the slope of our new line, we can use slope-intercept form and the given point to solve for the y-intercept.

$y=-\frac{3}{2}x+b$

$-1=-\frac{3}{2}(2)+b$

-1=-3+b

b=2

Plug the y-intercept into the slope-intercept equation to get the final answer.

$y=-\frac{3}{2}x+2$

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Example Question #1 : Parallel Lines

What is the equation of a line that is parallel to the line $\small y=\frac{1}{2}x+3$ and includes the point (4, 2) ?

Possible Answers:

$\small y=\frac{1}{2}x+6$

$\small y=-2x+10$

$\small y=2x-6$

$\small y=\frac{1}{2}x$

Correct answer:

$\small y=\frac{1}{2}x$

Explanation:

The line parallel to $\small y=\frac{1}{2}x+3$ must have a slope of $\frac{1}{2}$ , giving us the equation $\small y=\frac{1}{2}x+b$ . To solve for b, we can substitute the values for y and x.

$\small 2=(\frac{1}{2})(4)+b$

$\small 2=2+b$

$\small b=0$

Therefore, the equation of the line is $\small y=\frac{1}{2}x$ .

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Example Question #1 : Coordinate Geometry

What line is parallel to 5x + 3y = 8 , and passes through the point (6,-4) ?

Possible Answers:

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

$y = \frac{-2}{3}x + 5$

$y = \frac{3}{5}x - 2$

$y = \frac{-5}{3}x + 6$

$y = \frac{3}{2}x - 4$

Correct answer:

$y = \frac{-5}{3}x + 6$

Explanation:

Converting the given line to slope-intercept form we get the following equation:

$y = \frac{-5}{3}x + \frac{8}{3}$

For parallel lines, the slopes must be equal, so the slope of the new line must also be $\frac{-5}{3}$ . We can plug the new slope and the given point into the slope-intercept form to solve for the y-intercept of the new line.

y = mx + b

$-4 = \frac{-5}{3}(6) + b$

-4=-10+b

b = 6

Use the y-intercept in the slope-intercept equation to find the final answer.

$y = \frac{-5}{3}x + 6$

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

What line is parallel to 2x+3y=6 at (3, 2) ?

Possible Answers:

$y=-\frac{3}{2}x+8$

$y=-\frac{3}{2}x+4$

$y=-\frac{2}{3}x+4$

$y=-\frac{2}{3}x-4$

None of the answers are correct

Correct answer:

$y=-\frac{2}{3}x+4$

Explanation:

Find the slope of the given line: y=mx+b (slope intercept form)

$y=\frac{-2}{3}x+2$ therefore the slope is $\frac{-2}{3}$

Parallel lines have the same slope, so now we need to find the equation of a line with slope $\frac{-2}{3}$ and going through point (3, 2) by substituting values into the point-slope formula.

$2=\frac{-2}{3}(3)+b$

So, b=4

Thus, the new equation is $y=\frac{-2}{3}x+4$

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Example Question #1 : Parallel Lines

If the line through the points (5, –3) and (–2, p) is parallel to the line y = –2x – 3, what is the value of p ?

Possible Answers:

–10

–17

Correct answer:

Explanation:

Since the lines are parallel, the slopes must be the same. Therefore, (p+3) divided by (–2–5) must equal –2. 11 is the only choice that makes that equation true. This can be solved by setting up the equation and solving for p, or by plugging in the other answer choices for p.

$\frac{p--3}{-2-5}=\frac{p+3}{-2-5}=-2$

$\frac{p+3}{-7}=-2$

$(\frac{p+3}{-7})-7=-2(-7)$

p+3=14

p=11

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SAT Math : Coordinate Geometry

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #1 : Lines

Example Question #1 : Lines

Example Question #2 : Lines

Example Question #1 : Parallel Lines

Example Question #2 : Parallel Lines

Example Question #1 : How To Find The Equation Of Parallel Lines

Example Question #1 : Parallel Lines

Example Question #1 : Coordinate Geometry

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

Example Question #1 : Parallel Lines

All SAT Math Resources

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