In the xy-coordinate plane, the slope of line l is 3/4. Does line l pa

let the line be: y=mx+c

where m is the slope and c is the y intercept.

so y=(3/4)x+c

Statement 1:
if the line passes through 4,4 then 4=(3/4) 4 + c.. That implies c = 1. hence the equation of the line is y=(3/4)x+1. Now you can find whether line passes through -2/3,1/2

Similarly for statement 2.

You don't need to calculate all of this. You need to tell whether the statement is 'sufficient' to get an answer. in general, when the slope and a point is given you can find the equation of the line. Therefore, you can find whether any random point lies on that line or not.

Hence D.

In xy-coordinate plane, the slope of line L is 3/4. Does line L pass through point (-2/3, 1/2)?

(1) Line L passes through (4 ,4)
(2) Line L passes through (-4, -2)

as the per the problem statement -
y = mx + c
y = ( 3/4 )x + c

now plug in the point (-2/3, 1/2)
we get
1/2 = ( -2/3) ( 3/4) x + c
on solving the above we get
c = 0

hence the question becomes c = 0 ?

St 1 ==> Line L passes through (4 ,4)
the slope of line L is 3/4

now use the equation y = mx + c
plus the values ...m = 3/4 and x = 4 , y = 4

on the solving the above we get c =1
no ..definite ans . Sufficient .

St 2 ==> solve similarly as statement 1
we again get c = 1

sufficient

hence D ans

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If two unique points (x,y) and (p,q) are given for a line,the slope of this line = (q-y)/(p-x) or (y-q)/(x-p).

From F.S 1, we know that line passes through (4,4). Assume that it does pass through (-2/3,1/2). Then, the slope of the line has to be 3/4. Now the slope = (4-1/2)/[4-(-2/3)] = (7/2)/(14/3) = 3/4. Thus, it indeed passes through (-2/3,1/2). Sufficient.

From F.S 2, similarly, the slope = (-2-1/2)/[-4-(-2/3)] = (-5/2)/(-10/3) = 3/4. Thus, it passes through the given point. Sufficient.
Note that in case the slope didn't turn out to be 3/4, we would STILL be able to answer the question stem, with a YES or a NO.

D.
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NO need to calculate anything. If you know two points on a line, you can calculate the unique slope for that line. From both the Fact statements, we can arrive at a unique slope and answer the Question Stem.

D.
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Slope equation between two points is slope = y2-y1/x2-x1

We know that our line has a slope of 3/4. Each of the statements also gives us a point on the line. All we really need to do is find whether the slope bwtween the point given and (-2/3,1/2) is dequal to 3/4 or not.

stat. (1) 4-1/2 / 4-(-2/3) = 3.5 / 4+2/3 = 7/2 / 14/3 = 7/2 * 3/14 = 1/2 * 3/2 = 3/4. the answer is yes. Sufficient

Stat. (2) -2-1/2 / -4-(-2/3) = -2.5 / -4+2/3 = -5/2 / -10/3 = -5/2 * 3/-10 = 1/2 * 3/2 = 3/4. the answer is yes. Sufficient

the answer is D.

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

In xy-coordinate plane, the slope of line L is 3/4. Does line L pass through point (-2/3, 1/2)?

(1) Line L passes through (4 ,4)
(2) Line L passes through (-4, -2)

In the original condition, we only need to know the slop and the y-intercept, but the slope is given to be 3/4, so we only have one variable. This means we require only 1 equation when there are 2 equations given from the 2 conditions; there is high chance (D) will be our answer.
From condition 1, the answer can be either yes or no, so it is sufficient,
and in condition 2, the answer also can be both yes and no, so it is sufficient as well, making the answer D.

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.

Please note statement 1 has been changed from (4,4 ) to (-4, 4)

Well (-4,4) means line L does not pass through point (-2/3, 1/2)
so both the statements will contradict each other.

if statement 1 can be edited to the original format ( 4,4) that could help.
Thanks
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let the line be: y=mx+c

where m is the slope and c is the y intercept.

so y=(3/4)x+c

Statement 1:
if the line passes through 4,4 then 4=(3/4) 4 + c.. That implies c = 1. hence the equation of the line is y=(3/4)x+1. Now you can find whether line passes through -2/3,1/2

Similarly for statement 2.

Hence D.

This is a very easy question and you don't need any formula or anything to get the answer.

We know the slope of the line and want to know whether it passes through some point A.

(1) says that the line passes through some point B. Now, the slope shows the incline of a line. So, we know that the line passes through point B and know its incline, we can say whether it passes through A or not. Jut imagine, we have point B and know positioning of the line through it, we can answer any question about this line.

The same for (2).

Or, a slope of a line and one point the line passes through, are enough to get the equation of the line (check here: https://gmatclub.com/forum/math-coordin ... 87652.html), so from each of the statements we can get the equation and thus answer any question regarding that line.

Hope it helps.
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Hi BrentGMATPrepNow, when using slope formula, y = mx+c > 1/2 = 3/4x + c > c=0 ?
On St,1 , y = mx+c > 4 = 3/4 (4) + c > c=1 .
Therefore line l doesn't pass through the point (-2/3, 1/2)

However when using another slope formula = y2 -y1/ x2 -x1 in St 1> 4-1/2 / 4-(-2/3) > 3/4.
Therefore line l pass through the point (-2/3, 1/2)

My confusion here is why is different result and whether line l pass through the points (-2/3, 1/2) or not?
Thanks Brent

Hi Vinay,

Will you please explain with formulas or steps ?

Thanks in advance.

Equation of the line can be formed by using the slope and a point using the slope-point form of the line equation. Both A and B provides for the point needed to get the equation. Using equation of line we can check for the required condition. D wins.

Thanks for noticing.
+1.
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I did understand the explanation, but with difficulty.

Does someone knows where I can find good theory of line equations and slope formula and some excercises to practice with?

Thanks in advance

let the line be: y=mx+c

where m is the slope and c is the y intercept.

so y=(3/4)x+c

Statement 1:
if the line passes through 4,4 then 4=(3/4) 4 + c.. That implies c = 1. hence the equation of the line is y=(3/4)x+1. Now you can find whether line passes through -2/3,1/2

Similarly for statement 2.

Hence D.

The equation for a line is y=mx + b, where m equals the slope and b equals the y intercept. The slope is known, so the question being true or false will depend on the y intercept. To solve, simply find out what the y intercept would need to be to make the statement true and compare that to the value you get for S1 and S2. If the match the answer to the question is yes and if not it's no.

ANSWER: D

TheKingInTheNorth

When I plug in the point (-2/3, 1/2), I get c = 1, not c = 0. What am I doing wrong?

1/2 = (3/4)(-2/3)+c
1/2 = -6/12+c
1/2 = -1/2+c
Now add 1/2 to both sides
c = 1

Bunuel

I would be so appreciative if you could share how you would solve this problem. Is there a theory here to apply? Thank you for your help.

In the highlighted portion above, you took y = mx+c and replaced y with 1/2 but you didn't replace x with -2/3.

Having said that, I wouldn't bother finding any slopes or equations.

Instead, we can use the principles described in this video https://www.gmatprepnow.com/module/gmat ... /video/884

The given information locks in the slope of the line (it's 3/4).

Statement 1: Once we know that the line (with slope 3/4) passes through (4,4), the line is 100% locked in.
So we COULD draw the exact line and see whether or not it passes through (-2/3, 1/2)
This means statement one is sufficient

Statement 2: Apply the same reasoning.

Answer: D