GRE Algebra - Linear Equations

An equation is a relationship between two quantities, the quantity on the "right" side of the equal sign and the quantity on the "left" of the equal sign. It's a good chance that you'll see at least one linear relationship question the math section of the GRE exam. In a linear equation, each variable is NOT raised to a power. The following are all linear equations:

3 + x = 22
12y + 3z = 23 - z
x = 3
n + 3m = 22x

A linear equation can have many variables, or can have just one. Usually, the more variables there are, the harder the problem, but most questions on the GRE have only one or two variables. In all but the hardest problems, you'll be asked to solve for one of the variables, and this may require manipulating an equation in one of several ways. There are two general rules for manipulating equations, namely:

  1. You can add or subtract the same number from each side of the equation.
  2. You can multiply or divide each side of the equation by the same number.

Here is an example where you have to do minimal manipulation:

Step 0: Original Equation, solve for y 3 y + 2 = 15
Step 1: Subtract 2 from each side 3 y = 13
Step 2: Divide both sides by 3 y = 13/3

Here is a more detailed example, but notice that you are still using the same operations, just several of them:

Step 0: Original Equation, solve for x 2 x + 3 + 3 x = 28
Step 1: Simplify the left side by adding the xs 5 x + 3 = 28
Step 2: Subtract 3 from both sides5 x = 25
Step 3: Divide both sides by 5x = 5

It gets just a little bit harder when there are two variables in an equation and you are asked to solve for one of the variables "in terms of the other." In such a case, you need to remember that you can add, subtract, multiply, or divide both sides of the equation by the same number OR the same variable. For example:

Step 0: Original Equation, solve for s in terms of t 3s + 3t = 12
Step 1: Subtract t from both sides 3s = 12 - 3t
Step 2: Divide both sides by 3 s = 4 - t

Notice that in the last step in the previous example, you had to divide (12 - 3t) by 3, and you do that by dividing the quantities on the left and right of the equal sign by 3.