Lesson: Slope

Comment on Slope

Does a line with negative slope have to pass by quadrant 1? I just drew a line imaginary that did not pass through quadrant 1 and only 2,3,4>
gmat-admin's picture

A line with a negative slope will always pass through quadrants 2 and 4, but the line need not pass through quadrant 1 (the line COULD pass through quadrant 1, but that is not guaranteed).

For example, the line y = -2x - 1 does not pass through quadrant 1.

Love the concept of rise/run to explain slope easy and elegant ! Thank you.

Hey, love your videos, keep doing the doing the great work. My question is, if we compare two increasing or decreasing lines, can we know for sure whose slope is less or more. How can i obtain a general understanding for every quadrant?
gmat-admin's picture

Good question!

The steeper the slope, the greater the MAGNITUDE of the line.

So, for example, if the two slopes are both POSITIVE, then the slope of the steeper line will be greater than the slope of the line that is less steep.
For example, a line with slope 4 will appear steeper than a line with slope 2.

If the two slopes are both NEGATIVE, then the slope of the steeper line will be less than the slope of the line that is less steep.
For example, -4 < -2. So, a line with slope -4 will appear more steep than a line with slope -2.

Does that help?

PS: Glad you like the videos!

please explain
gmat-admin's picture

Some of the questions on the DS part of the Quant ask to find if the line passes through a certain Quadrant. I'm wondering if there's a general rule in which slopes pass through one or more quadrants based on the value of the slope (so, if slope is -1/2 it goes through Quadrant III and V, slope 3 at Quadrants I, II, III etc etc.).
gmat-admin's picture

It would be hard to generate a concise rule that covers all possibilities.
The reason for this is that, if we take two lines with the same slope and move them up or down, then we can change the quadrants they pass through.

For example, the lines y = 2x + 5 and y = 2x - 5 have the same slope.

However, the line 2x + 5 passes through quadrants I, II and III
And, the line 2x - 5 passes through quadrants I, III and IV


If line M passes through the coordinate system shown above, does line M pass through Quadrant IV?

(1) Line M has a slope greater than 1/2.
(2) Line M has a slope greater than -1.

How do I test if either is sufficient?
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-line-m-passes-through-the-coordinate-syste...

For each statement, identify a slope that meets the condition.
So, for example, for statement 1, the slope COULD equal 2.
On a piece of paper, draw the x-y coordinate plane and sketch a line with slope 2.
As yourself "Does this pass through Quadrant IV?"
If it DOES pass through Quadrant IV, ask yourself "Can I draw another line with slope 2 so that is does NOT pass through Quadrant IV?"
If you can do that, then you have shown that statement 1 does not help us answer the target question with certainty.
As such, the statement is not sufficient.

In my solution (link below), you'll see that my two diagrams satisfy each statement on its own, AND they also satisfy the statements combined.

Does that help?

Here's my solution: https://gmatclub.com/forum/if-line-m-passes-through-the-coordinate-syste...


Maybe I missed this guide in the videos, but how do you roughly plot the slope of a line without any coordinates to see if the slope touches any of the quadrants?
gmat-admin's picture

Your sketch doesn't need to be very precise to answer the question. In fact, a very crude sketch will still yield the correct answer.

That said, if you want to sketch a line with slope 2, then this means the rise/run = 2.
This is the same as the rise/run = 2/1.
So for every 2 units you rise (go up), you need to travel 1 units in the positive horizontal direction.

This strategy is covered in the first 2 minutes of the above video.

Does that help?


That helps. Am I right in saying that if we need to determine exactly which Quadrants slope passes through (with the exception of a slope being undefined and / or equal to 0) then we need to know whether the line is below or above the x / y intercepts?
gmat-admin's picture

I think that's true.
To be sure, here's what we can conclude about which Quadrants a line passes through:

1) Positive slope and positive y-intercept: I, II and III
2) Positive slope and negative y-intercept: I, III and IV
3) Negative slope and positive y-intercept: I, II and IV
4) Negative slope and negative y-intercept: II, III and IV


Hello Brent,

Could you elaborate on what the slop of a line would be like if it is perfectly horizontal or vertical?
gmat-admin's picture

Good question!

A HORIZONTAL line has slope ZERO.
As we travel from left to right from one point on the line to another point on the line, the RISE = 0, and the RUN = some number.
So, slope = rise/run = 0/some number = 0

A VERTICAL line has an UNDEFINED slope.
As we travel from one point on the line to another point on the line, the RISE = some number, and the RUN = 0.
So, slope = rise/run = some number/0 = undefined


Hi Brent,

Could you please explain how to find the Y- Co ordinate in this question ?


Please explain me the laws of sines . Thankyou !
gmat-admin's picture

Here's my full solution: https://gmatclub.com/forum/in-the-figure-above-what-are-the-coordinates-...

The Laws of Sines is beyond the scope of the GMAT.
However, if you're interested, check out this page: https://www.mathsisfun.com/algebra/trig-sine-law.html



Hi Brent I think you meant to put option E :) isn't it (5-5). Thank you!
gmat-admin's picture

Good catch!
I've edited my response.

Hi Brent,

I saw your answer to this question on GMATCLUB, yet I don't understand why you wrote that the slope of the line n is 2 and not 1/2


Let me use an analogy:

Let's take the graph y=x, it has a slope 1
Let's take the graph y=-x, it has a slope -1

Those two lines are perpendicular, therefore

y=-(1/2) * x must be perpendicular to y=1/2 * x

Fortunately, It does not have any effect on DS question type, yet I am afraid that If it were a PS question I would answer it incorrectly, that is why I want to clarify this.

Thank you in advance,
gmat-admin's picture

Key property: If line L is perpendicular to line M, then the slope of line L is the negative reciprocal up the slope of line M.

So, for example, it could be the case that line L has slope 2/3, which means line M must have slope -3/2


Hi Brent,


I found this video on Khan Academy, Now I understand why the slope of one line is negative reciprocal of a line that is perpendicular to it.

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