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

Notice that line l passes through points (3,0) and (0,2), so its slope is \(m=\frac{y_2-y_1}{x_2-x_1}=-\frac{2}{3}\) (given two points \((x_1,y_1)\) and \((x_2,y_2)\) on a line, the slope \(m\) of the line is \(m=\frac{y_2-y_1}{x_2-x_1}\)).

Only option B, when written in \(y=mx+b\) form has the slope of -2/3.

Answer: B.
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Hi Bunnel...A small query
Can we assume (As u have done in your solution) that the line passes through points (3,0) and (0,2). I mean can we safely interpret graphs on the GMAT??Since it is not explicitly stated that the line passes through those two specific points.
My approach to the above problem was as follows.
What we know from the graph is this
x and y co-ordinates of line l are both positive, x co-ordinate is > 2 and y co-ordinate is > 1
Checking answer choices -
A, D and E are clearly out since x and y co-ordinates will have opposite signs
putting y=0 in choice C x=2 - Incorrect (as x co-ordinate > 2)
Only B left
Cheers

OG13 solution makes the same exact assumption "The line is shown going through the points (0,2) and (3,0)..."
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From the given figure, we can see that the line has positive intercept on both the x and y axis. Thus we can eliminate all the options except B and C. Now the intercept on the x-axis is more than 2. For option C, the x-intercept comes as 2, thus the answer has to be B.
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Just a different way of approaching the problem...
since we know the y-intercept is -c/b when the equation is given in the form ax+by +c =0.. simply use what's on the right side of the equation (it'll turn negative when moved to the left) and divide by b.

I agree with Dipankar6435 approach. I used the similar approach since it is not explicitly mentioned that the line passes through and also there is no line markers at the points where the line touches both x and y axis. Since there is a line marker on the x axis that clearly indicates that the x value should be greater than 2, out of the 2 answer choices, I picked B as the right one. Thanks.

Very very strange because in Maths , one is groomed to never make assumptions based on diagrams unless explicitly mentioned .

There is another general form of line in co-ordinate plane which is:


x/a + y/b = 1

where a is the point of intersection of line and x-axis and b is the point of intersection of line with y-axis.
Here a=3, b=2
Therefore,

x/3 + y/2 = 1

or

2x + 3y = 6

This solution is valid, if we assume the values of a, and b.

But even if we don’t assume these values, we can eliminate option A), D), and E) because, we can see that both the x intercept and y intercept are positive.

Now, we see a>b through observation, which means coefficient of x is greater than coefficient of y, which is in option B) only. Hence B is the answer.

If intercept on the x axis and y axis is known, intercept formula for line is the fastest method to get the equation of the line.

Equation of a line which cut an intercept a at x axis and b at y axis is given by

(x/a) + (y/b) = 1

However, a more useful form of this equation is

bx + ay = ab

using this, equation of line can be found easily by inspection only
Here , Intercept at x axis = a = 3
Intercept at y axis = b = 2
hence, equation of line by putting values
2x+3y =6 hence answer is B

L = (0,2)
X = (3,0)

slope = 2-0/0-3 = -2/3

y = mx + b
y = -2/3x + b

Find b:

2=0+b
b = 2

y=-2/3x + 2
x 3 to eliminated -2/3

3y = -2x +6
3y + 2x = 6

Without any calculation Approach!
We know from the graph that X-intercept must be greater than 2.
So, co-efficient of x must be 2. Hence C and E are out.
The equation of straight line is y=mx+b and In the figure we can see that slope must be negative.
Therefore, A and D are out since they have positive slope.
Answer is B.

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 line l has two points: (0,2) and (3,0). 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. This gives us:

3y = -2x + 6

Then add 2x to both sides of the equation:

2x + 3y = 6

The answer is B.
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We can see that points (0,2) and (3,0) are ON THE LINE. So, their coordinates must SATISFY the equation of the line.

Let's start with (0,2).
(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

Answer:

RELATED VIDEO

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For this problem, I find it easier to use answer choices. Only one answer choice can work for both coordinates.

Let's say P1 (0; 2) and P2 (3; 0)
for P1 x=0, y=2
for P2 x=3, y=2
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Using P1 coordinates:
(a) 2*0 - 3*2 = - 6 , not sufficient
(b) 2*0+3*2= 6 - sufficient
(c) 3*0 + 2*3 = 4, not sufficient
(d) 3*0 - 3*2 = -6, also sufficient
(e) 3*0 - 2*2 = -4, not sufficint

So we have 2 answers that are sufficient for P1, b and d.

Using P2 coordinates
(a) 2*3 - 3*0 = 6, sufficient
(b) 2*3 + 3.0 = 6, also sufficient
(c) 3*3 +2*0 = 9 - not sufficient
(d) 2*3 - 3*0 = 6 - not sufficient
(e) 2*3 - 2*y = 6, not sufficient.

Here we also have 2 answers that are sufficient, but only 1 answer is sufficient for both coordinates, which is B.

A very simple approach would be -
Notice line l has a negative slope and the line l intersects in 0,2 and 3,0. Only option B & C satisfies i.e they have a negative slope { rewrite them in slope intercept form y= mx + b}

B. y = -2/3x + 2 { put the points 0,2 & 3,0 - they both satisfies the equation}
C. y = -3/2x + 2 { put the points 0,2 & 3,0- notice 3,0 doesn't satisfies the equation}.


Hope you understood.
Please press the kudos for the appreciation.

Thanks

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Equation of a line can be writter as (x/a) + (y/b) =1
where a = x intercept & b=y intercept
from figure both a & b are + ve => only options B & C remain

And again from figure a>b => option B is the best possible choice

From the figure we know that slope is negative ..only option B has negative slope

study mode

hi

since x and y intercepts are given, lets try the below method

(x/a) + (y/b) = 1 [ where a is x intercept and b is y intercept ]

which implies

2x +3y = 6 = answer choice B

hope this helps!
cheers and thanks!

I think that first you should eliminate the options who don't have negative slope. Your process is tedious if you don't do that before. Hope I was clear.