Solving GMAT Questions with two linear equations and two unknowns

In order to solve such equations, you need at least 2 distinct equations involving these unknowns.

For example, if we are trying to solve for x and y, we won't be able to solve it using these 2 equations.

2x + y = 14
4x + y - 14 = 14 - y

Why?  Because the two equations on top are the same.  If you simplify the second equation, you get 4x + 2y = 28 which reduces to 2x + y = 14 - the same equation as the first.  If the two equations are the same, then there will be infinitely many values for x and y that will satisfy the equations.  For example, x = 2 and y = 10 satisfies the equation.  So does x = 4 and y = 8.  And so does x = 6 and y = 2.

In order to solve for an actual value of x and y, we need 2 distinct equations.

For example, if we had

2x + y = 14 --------(1)
x - y = 4 ----------(2)

Then from equation (2), we can get x = 4 + y and substitute that into equation (1) to get:
2(4 + y) + y = 14  We can then solve for y.  See if you got y = 2  Once you've got y = 2, you can substitute that into x= 4 + y to get x = 6.

An important lesson here is that you need as many distinct equations as you have variables.  So if you are doing a data sufficiency question, you can sometimes just note that as long as you have two distinct equations, you will be able to solve for x and y.  Disregard or other equivalent equations.

Sometimes, however, its possible for there to be no solution to the set of equations.  This occurs when one side of each equation is the same, but the other side is different.  For example,

2x + y = 14
2x + y = 0

has no solution.  Its like saying that the same steak costs $14 and $0 at the same restaurant.  Of course, the GMAT is not going to make it so obvious that the equations contradict each other. 

Usually, you would have to simplify one of the equations to check if it is

1. the same same as the other equation, in which case, you have infinitely many solutions
2. the same as the other equation on one side, but different on the other side of the equal sign.  In this case, you have no solution
3. a distinct equation, in which case x and y has an exact value.

The GMAT might also require you to translate a word problem into a pair of simultaneous equations to solve.  Let's try this question from Grockit.

A package contains nothing but 35 DVDs and 15 videotapes. What is the total weight, in pounds, of the contents of the package?

(1) Each videotape weighs twice as much as each DVD.
If we let the weight of a DVD be x and a videotape be y, then according to this statement y = 2x
(2) The total weight of 2 of the videotapes and 2 of the DVDs is 1 pound.

Following the previous notation, 2y + 2x = 1