In a rectangular coordinate system, straight line k passes through poi

slope of line ; 2/3
for each given option IMO E has slope = 2/3 it means that it lies on the same line
IMO E

Okay here is my approach:

Most of the time with coordinate questions, when you see two points given, you most likely will need to calculate the slope.

Slope of line = (Y1-Y2)/(X1-X2)

Point 1: (0,0) therefore X1= 0, Y1 = 0

Point 2: (3,2) therefore X2=3, Y2=3

Plug into slope formula: (2-0)/3-0) = 2/3

Now that we know the slope, the next step is to incorporate the slope into a y =mx + c format.

So we get: y=(2/3)x + c

If we take any of the points i.e. point 1 or 3 and plug it into y=(2/3) + c you get that the c = 0. So the equation of the line is y=(2/3)x

Now looking at the answer options, plug in each answer options into y=(2/3)(x). The goal is to see if both side of equations equal each other. That would be the answer.

When substituting answer options, I've read and learned that you should start with option C, which is what I did. Substituting the answer choices into y=2/3x is relatively quick. You will fine that the answer is E i.e. :

y=(2/3)x

Point given: (-6, -4). Substitute this into the equation:

-4 = (2/3) (-6)

-4 = -4

Answer: E

Tips to remember:
Always find slope when two points are given
When substituting answer choice, start with C and work your way up or down from there. It is said that if you do that, there's a higher chance that you can find the answer quickly and save time (but this is of course, not always the case). Also, I read somewhere that if the answer ask "which is the following ...." in the question, then the answer is most likely D or E. I don't know how true this is (and I don't think it is always right) but we need all the strategy we can get right ? Give it a shot if you are running out of time!

(y-0)/(x-0) = (2-0)/(3-0)

3y = 2x, only E. (–6, –4) satisfies.

hi brent, can you please advise how to choose C=o, or what is the theory to find C/ y intercept in case of 2 points given,
I solved this question by 3x=2y theory, but i wan to know how we solve by line theory

You might want to review the slope y-intercept format of expression equations of lines. I'd say the GMAT test-makers are partial to that format.
That is, y = mx + b, where m = the slope and b = the y-intercept.

Since we're told line k passes through the point (0,0), we can see that line k "intercepts" the y-axis at y = 0.
This is how we know the y-intercept is 0

Does that help?

Cheers,
Brent

Someone please draw the co ordinate figure

Posted from my mobile device

but there are 2 points given, for point (3,2), the y int will be 2?

Since line k passes through points (0, 0) and (3, 2), its slope is (2 - 0) / (3 - 0) = 2/3. If a point is on line k, then the slope calculated using its coordinates and one of the given points should be 2/3, the slope of the line. Since (0, 0) is an “easier” point for calculations than (3, 2), we will use (0, 0). For choice A, the slope between (0, 0) and (9, 4) is (4 - 0) / (9 - 0) = 4/9. We see that this is not 2/3, so choice A is not the correct answer.

Notice that the slope between (0, 0) and (a, b) is b/a. Therefore, we can determine the slope of the line easily for the other four choices and determine which is on line k.

B. (4, 9) → slope = 9/4 ≠ 2/3

C. (-4, 6) → slope = 6/(-4) = -3/2 ≠ 2/3

D. (-6, -9) → slope = (-9)/(-6) = 3/2 ≠ 2/3

E. (-6, -4) → slope = (-4)/(-6) = 2/3

Answer: E

hey brent,

If i take the coordinates (3,2) Can i solve assuming Y intercept as 2. Please help

No. The given information already tells us (indirectly) that the the line passes through the origin (0, 0).
If the line passes through (0, 0), then the x-intercept is 0, and the y-intercept is 0.

Here a lesson from Khan Academy that might help:
https://www.khanacademy.org/math/algebr ... from-graph

The equation (slope) \(y =\) \(\frac{0-2}{0-3}\)

\(y =\) \(\frac{2}{3}\)x

3y = 2x
-2x+3y=0

I any points satisfy the equation the point will be on the line. We need to try every option. Only E satisfies the equation.

-2*(-6) +3(-4)=0
12-12=0
0=0

So E is the answer.

BrentGMATPrepNow

Hi Brent,


how can I know that the y-intercept is 0 and not 2 ?

Many thanks

The y-intercept is the point on the graph (line) that crosses the y-axis.
Every point on the y-axis has 0 as its x-coordinate.
So for example, if a line crosses the y-axis at the point (0, 7), then the line has y-intercept 7.
Similarly, if a line crosses the y-axis at the point (0, -5), then the line has y-intercept -5.

Since we're told the line passes through the point (0,0), we know that the line crosses the y-axis at the point (0,0), which means the line has y-intercept 0

BrentGMATPrepNow

thanks, so we should only consider the first one that they provide? why we do not consider (3, 2) ?



Many thanks

Every line is comprised of infinitely many points.
For this question, we are provided with two of those points.
One of those points (0,0) lies on the y-axis, and the other point (3,2) doesn't lie on the y-axis.
The point that lies on the y-axis tells us what the y-intercept is.
The point that doesn't lie on the y-axis doesn't tells us anything about the y-intercept.

Not necessary, once you find the slope you will get the equation y = 2/3x + C, once you have got this equation substitute any of the points into the equation, y intercept will be the value when x = 0. Hence y = C.

Now if you take (x,y) = (3,2) => 2 = 2/3*(3) + C => 2=2+C Hence C = 0

If you take 0,0 also you will get the same C = 0

Slope of given line will y2-y1/x2-x1

1 coordinate mentioned (3,2), 2nd one mentioned (0,0) Slope of line will be 2-0/3-0 = 2/3,
for the any coordinate on line its slope wrt other point should be 2/3, we can consider (0,0) as another point, then essentially slope will be y/x, now quickly check for all options

A. (9, 4) = 4/9 - not equal to 2/3
B. (4, 9) = 9/4 - not equal to 2/3
C. (–4, 6) = 6/-4 = -3/2 not equal to 2/3
D. (–6, –9) = -9/-6 = 3/2 not equal to 2/3
E. (–6, –4) = -4/-6 = 2/3 right answer