GmatKiss wrote:In the rectangular coordinate system, a line passes through the points (0,5) and (7,0). Which of the following points must the line also pass through?
(-14, 10)
(-7, 5)
(12, -4)
(14, -5)
(21, -9)
Hi!
Let's start with what's probably the quickest approach: actually graphing the line.
The notepad that you get on the GMAT is graph-lined, so it's pretty easy to accurately draw out coordinate geometry questions. Accordingly, one option for solving this problem is to plot the two points, draw the line and see which answer also falls on the line.
This approach wouldn't get you 10/10 on a high school geometry test, but on Test Day we couldn't care less about elegant solutions - we're far more concerned with efficient ones.
Of course, there's also the traditional approach: use the two points to find the equation of the line and then plug each answer into the equation until you find the one that fits.
First we solve for the slope of the line. Since slope = rise/run = (change in y)/(change in x), we get:
slope = (5-0)/(0-7) = 5/-7 = -(5/7)
Now we know that our equation is:
y = -(5/7)x + b
To solve for b, we plug in one of our two points. Letting x=0 will make the math way simpler, so let's plug in (0,5):
5 = -(5/7)*0 + b
5 = b
Now we have our full equation:
y = -(5/7)x + 5
At this point we can plug in the choices until we find a match:
a) -(5/7)(-14) + 5 = 10 + 5 - 15.. bzz
b) -(5/7)(-7) + 5 = 5 + 5 = 10... bzz
c) -(5/7)(12) + 5 = not an integer... bzz
d) -(5/7)(14) + 5 = -10 + 5 = -5.. ding ding ding! Choose (d)!
As an aside, here's an interesting statistical anomaly on problem solving questions that include the phrase "which of the following": the answer is (d) or (e) MORE than 40% of the time. So, when you see that phrase in a problem solving question and you have to work with the answers, you gain a slight edge by starting at the bottom and working your way up!
