GMAT Data Sufficiency: Systems of Equations

Most of you are probably familiar with the basic rules for solving linear equations or systems of linear equations, yet many GMAT takers fall prey to booby traps planted in these kinds of Data Sufficiency questions. In order to avoid such mistakes, you should learn well the general methods for solving Systems of Equations and become acquainted with the special cases that may appear on your test.

A set of equations to be solved simultaneously is called a System of Equations. The general rule is that a linear system with n variables usually needs n independent equations to solve it. However, the GMAT Maths questions sometimes uses special cases—offering dependent equations or asking for the value of an expression—that make things a bit more complicated. Let’s examine these more closely.

Beware of DEPENDENT equations

Sometimes an equation in a system does not add essential information but just repeats information already presented by other equations in the system. Such an equation, which is called dependent, is useless and can be eliminated from consideration. Here’s an example:

2x + y + 3z = 1
3x + z = 1
x – y – 2z = 0

If you subtract the first equation from the second, you get:

3x + z – (2x + y + 3z) = 1 – 1
3x + z – 2x – y – 3z = 0
x – y – 2z = 0, the third equation.

This third equation is dependent, because it merely restates the difference between the first two equations. Once this dependent equation is eliminated, the system transforms:

2x + y + 3z = 1
3x + z = 1

Now the number of variables (three) exceeds the number of equations (two), which means that you cannot solve this system to find unique individual values for x, y, and z.