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## Find the equation of the line

tagged by: Brent@GMATPrepNow

This topic has 4 expert replies and 1 member reply
GmatGreen Senior | Next Rank: 100 Posts
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85 messages

#### Find the equation of the line

Thu Mar 13, 2014 12:28 pm

In the coordinate system above, which of the following is the equation of line l?

A) 2x - 3y = 6
B) 2x + 3y = 6
C) 3x + 2y = 6
D) 2x - 3y = -6
E) 3x - 2y = -6

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Brent@GMATPrepNow GMAT Instructor
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Wed Aug 23, 2017 4:53 pm
GmatGreen wrote:

In the coordinate system above, which of the following is the equation of line l?
A) 2x - 3y = 6
B) 2x + 3y = 6
C) 3x + 2y = 6
D) 2x - 3y = -6
E) 3x - 2y = -6
We can see that points (0,2) and (3,0) are ON THE LINE. So, their coordinates must SATISFY the equation of the line.

(A) 2x - 3y = 6. 2(0) - 3(2) = -6 ELIMINATE
(B) 2x + 3y = 6. 2(0) + 3(2) = 6 KEEP
(C) 3x + 2y = 6. 3(0) +2(2) = 4 ELIMINATE
(D) 2x - 3y = -6. 2(0) - 3(2) = -6 KEEP
(E) 3x - 2y = -6. 3(0) - 2(2) = -4 ELIMINATE

Great. We're down to B or D

Let's test (3,0).
(B) 2x + 3y = 6. 2(3) + 3(0) = 6 KEEP
(D) 2x - 3y = -6. 2(3) - 3(0) = 6 ELIMINATE

Cheers,
Brent

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nikhilgmat31 Legendary Member
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Tue Jul 28, 2015 11:08 pm
simple way to find the equation with same slope & eliminate others.

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Patrick_GMATFix GMAT Instructor
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Thu Mar 13, 2014 12:40 pm
Equation of the line is y=mx+b where m is the slope and b is the y-intercept. In this case, the slope (change in y / change in x) is -2/3 (calculated from the two known points). The y-intercept (where the line crosses the y-axis is 2. So the equation of the line is y = (-2/3)x + 2. Manipulate it to put x's and y's on the same side and you'll get 2x+3y = 6.

The answer is C. I go through the question in detail in the full solution below (taken from the GMATFix App).

-Patrick

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GMATGuruNY GMAT Instructor
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Thu Mar 13, 2014 3:45 pm
GmatGreen wrote:

In the coordinate system above, which of the following is the equation of line l?

A) 2x - 3y = 6
B) 2x + 3y = 6
C) 3x + 2y = 6
D) 2x - 3y = -6
E) 3x - 2y = -6
An alternate approach is to PLUG IN THE ANSWERS.
The figure indicates that the following points are on line L: (0,2) and (3,0).
Since a line is defined by two points, only one answer choice can work for both (0,2) and (3,0).

Test (0,2) in the answer choices:
A) 2x - 3y = 6
2*0 - 3*2 = 6.
-6 = 6.
Doesn't work.
Eliminate A.

B) 2x + 3y = 6
2*0 + 3*2 = 6
6 = 6.
This works.

Since (0,2) works in B, test whether (3,0) also works in B:
2*3 + 3*0 = 6
6 = 6.
Success!

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Scott@TargetTestPrep GMAT Instructor
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Tue Jul 28, 2015 1:43 pm
GmatGreen wrote:

In the coordinate system above, which of the following is the equation of line l?

A) 2x - 3y = 6
B) 2x + 3y = 6
C) 3x + 2y = 6
D) 2x - 3y = -6
E) 3x - 2y = -6
We start by defining the equation of line l using the slope-intercept form of a line (y = mx + b), where m = slope and b = the y-intercept.

Notice that the two points (0,2) and (3,0) are on line l. We can use these two points to determine the slope. The formula for slope is:

m = (change in y)/(change in x) or

m = (y_2 - y_1)/(x_2 - x_1)

Plugging in our points we have:

m = (0 - 2)/(3 - 0)

m = -2/3

We also see from the diagram that the y-intercept of line l is 2. Substituting the slope and the y-intercept into the line equation we have:

y = (-2/3)x + 2

The final step is to recognize that the answer choices are in a different form than is our equation for line l. Thus, we have to manipulate our equation such that it will match one of the answer choices. Let's first multiply the entire equation by 3. Doing so gives us:

3y = -2x + 6

Then add 2x to both sides of the equation:

2x + 3y = 6

Alternate solution:

To obtain an equation of a line, we can also use the two-intercept form of a line (x/a + y/b = 1) where a is the x-intercept and b is the y-intercept of the line. This is a lesser-known form of an equation of a line, but it comes in handy when we know or are given the x- and y-intercepts of the line. Here, we see that the x-intercept of the line is 3 and the y-intercept is 2. Thus the equation in the two-intercept form is:

x/3 + y/2 = 1

Multiply the entire equation by 6, and we have:

2x + 3y = 6.

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