Quarter Wit, Quarter Wisdom: Bagging the Graphs - Part II

Quarter Wit, Quarter WisdomIf you haven’t read part I of this topic, I strongly suggest you read it first: Bagging the Graphs – Part I

Would you say it is easy breezy to draw a line if two points through which it passes are given? Sure. Plot the points approximately and join them! There you have your required line.

A quick method of drawing the line represented by an equation: find two points through which it passes, plot the points and join them.

The easiest points to find are x and y intercepts. Let’s say I have the equation 2x + y – 4 = 0. In this equation, if I put x = 0, I get y = 4 (the y intercept). This means the point (0, 4) lies on the line. If I put y = 0, I get x = 2 (the x intercept). So the point (2, 0) lies on the line too. Plot them and join them.

Are you wondering why I worked with the slope first when this easy method existed? Because slope method has its own utility and will come in very handy in some questions. Anyway, I had a nagging feeling that if I suggested this approach first, you may not bother about the slope concept at all!

One data sufficiency question to bind it all:

Question: Every point in the xy plane satisfying the condition ax + by ≥ c is said to be in region R. If a, b and c are real numbers, does any point of region R lie in the third quadrant?

Statements:

  1. Slope of the line represented by ax + by – c = 0 is 2.
  2. The line represented by ax + by – c = 0 passes through (-3, 0).

Solution:

First of all, notice that ax + by – c = 0 or ax + by = c is the equation of the same line. A line divides the plane into two regions. One of them, where every point (x, y) satisfies ax + by ≥ c, is region R. As of now, we do not know what the line looks like and which of the given two regions is region R.



If the line is as shown above, both the regions will have points in the first, second and fourth quadrants but only Region 1 will have points in the third quadrant.

But if the line is as shown in this figure, both the regions will have points in the second, third and fourth quadrants but only Region 2 will have points in the first quadrant. There are many different cases. We see that if a line passes through a particular quadrant, both regions lie in that quadrant. Else, only one of the two regions lies in that quadrant. For example, in the figure above, the line passes through second, third and fourth quadrants and hence, both regions lie in these quadrants. But it doesn’t pass through quadrant 1 and hence only Region 2 lies in the first quadrant. Let’s go on to the statements now.

Statement 1: Slope of the line represented by ax + by – c = 0 is 2.

If the slope of the line is 2, it will always pass through the third quadrant and hence both regions will have points in the third quadrant. (Note that a line with a positive slope will definitely pass through the first and the third quadrants.) Statement 1 is sufficient alone.

Statement 2: The line passes through (-3, 0).



A line passing through (-3, 0) could be the blue line or the green line. In either case, the line will pass through the third quadrant and hence, will have both regions in the third quadrant. So it is sufficient too? What about the x axis? That is also a line passing through (-3, 0). But it does not pass through the third quadrant. We would need the equation of the line to find out whether our region R lies in the third quadrant. The equation of x axis is y = 0. So the required region R is given by y ≥ 0 i.e. the first and second quadrants. Here region R does not have any points in the third quadrant. Using just the information given in Statement 2, we cannot say whether a point of region R lies in the third quadrant or not. Statement 2 alone is not sufficient.

Since statement 1 alone is sufficient and statement 2 alone is not, answer is (A).

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches GMAT prep courses for Veritas Prep in Detroit, Michigan, and regularly participates in content development projects such as this blog!

8 Responses

  1. Asha says:

    What i understand is that

    Statement 2 is not sufficient because:

    when x = -3. we get two situations:

    If y is less than or greater than 0 that the line will pass through the 3rd quadrant

    If y = 0, than the line will not pass through quadrant 3

    am i right?

    • Karishma says:

      Yes, you can say that if the y intercept of the line is less than or greater than 0, then the line will pass through the third quadrant. On the other hand, if the y intercept is 0 (i.e. the line lies on the x axis), it will not pass through the third quadrant

  2. Bala says:

    In the above example, it is said that “If the slope of the line is 2, it will always pass through the third quadrant and hence both regions will have points in the third quadrant.”… how did you figure this out…

    • Karishma says:

      The line divides the plane into two regions: region 1 and region 2I (shown by red and green lines in the diagram above). If the line passes through a quadrant, on its one side will be region 1 and on the other, region 2.

      If the slope of a line is 2, it must pass through the first and third quadrants. Think about it: it is a line, not a line segment. A line is infinite at both ends. At some point, the line will cross over to the third quadrant even if you try to make it in the first and fourth quadrants. If the line passes through the origin, it will not cross the second and fourth quadrant but in any case, it has to lie in the first and third quadrants. Try making lines of slope 2 which do not lie int he third quadrant – you can’t. It is a line. If you extend it enough, it will lie in the third quadrant.

  3. Jakelong says:

    Hi Karishma,

    Here I have a doubt, for the statement 1, only slope is given and the problem says that a,b,c are real numbers, so according to this, for statement 1, x can also take the value of 0 in which case the region will not fall in the third region.

    So with this line of reasoning we can also say that A is insufficient,

    Is my reasoning method right??

    • Karishma says:

      If slope is given to be 2, the equation of the line becomes
      y = 2x + c
      where c is the y intercept.
      When x = 0, y = c will be a point on this line. The line will be a slanting line passing through the first and the third quadrant for sure. If the y intercept is non-zero, it will also pass through the second/fourth quadrant.

  4. Mahmood says:

    ” The equation of x axis is y = 0. So the required region R is given by y ≥ 0 ”

    I didnt understand how you found the region R, y ≥ 0..The region is given by, ax + by ≥ c.. from here, how can we deduce what you wrote?

    Thanks a lot for the help.

    • Karishma says:

      Here the point is what if a = 0, b = 1 and c = 0? In that case the line ax + by – c = 0 becomes y = 0 which passes through (-3, 0). So these values satisfy statement 2.
      The region R given by ax + by >= c becomes y >= 0. In which region is y positive or 0? The first and second quadrant only. So in this case R has no points in the third quadrant.
      So in some cases in statement 2, R will have points in third quadrant and in this case it will not. So statement 2 alone is not sufficient.

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