GMAT Coordinate Geometry
August 1, 2012
The key to many GMAT coordinate geometry questions is to remember that coordinate geometry is just another way of expressing the possible solutions to a two variable equation. Each point on the line in a coordinate plane corresponds to a solution for the equation of that line.
The base equation for a line is y = mx + b, where b is the y intercept, or the point at which the line crosses the y-axis, and m is the slope, or the steepness of the line. More specifically, the slope of a line is the change in the y coordinates divided by the change in the x coordinates between any two points on the line.
While understanding the basic format for an equation of a line can be very useful on the GMAT quantitative section, you will encounter GMAT problems in which it is faster and easier to think of the problem in algebraic terms. In such cases you should think of the equation as an algorithm that will produce the y value given any x value. This is the reason that the x values are sometimes referred to as inputs and the y values as outputs.
For example, if your answer choices are solution sets and you are asked to determine which option is on the line given in the y = mx + b form, rather than graphing the line and trying to determine which point falls on it, which is especially difficult as you will not have graph paper, you can plug each x value into the equation and determine which one produces the appropriate y value.
On test day, the key is to remember that coordinate geometry is just a way of expressing algebraic concepts visually. Thus, we can often treat these problems as algebra rather than as geometry. To see this in action, try the problem below.
In the xy-coordinate system, if (m, n) and (m 1 2, n 1 k) are two points on the line
with the equation x 5 2y 1 5, then k 5
Step 1: Analyze the Question
For any question involving the equation of a line, a good
place to start is the slope-intercept form of the line,
y = mx 1 b. Remember that if you have two points on a
line, you can derive the entire equation, and if you have an
equation of the line, you can calculate any points on that
Step 2: State the Task
We are solving for k, which is the amount by which the
y-coordinate increases when the x-coordinate increases
Step 3: Approach Strategically
The slope of a line is the ratio between the change in y and
the change in x. In other words, every time the x-coordinate
increases by 1, the y-coordinate increases by the amount
of the slope.
The equation of the line in the question stem is defined as
x = 2y + 5. We must isolate y to have slope-intercept form:
So the slope of this line is 1/2 . This means that for every
change of +1 in the x direction, there is a change of + 1/2
in the y direction. Then we know that, because there is an
increase in 2 units in the x direction when moving from
m to m + 2, there must be a change of 1 unit in the y
direction when moving from n to n + k. So k = 1.
Since there are variables that eventually cancel (m and n
are not part of the answers), we can Pick Numbers. Let’s
say that you choose the y-coordinate of the point (m, n) to
be 0 to allow for easier calculations. Using the equation
we’re given to relate x- and y-coordinates, we can calculate
So (m, n) is the point (5, 0).
Now we’ll plug our values of m and n into the next point:
(m + 2, n + k). That yields (7, k). All we have to do is plug
an x-coordinate of 7 into the equation to solve for k, the