GRE Exam: A Simple Guide To Algebra - Linear Equations

In broad terms, you can think of algebra as any situation where you manipulate an equation or inequality to solve for one or more unknown values.

An unknown value is a variable (such as x or y) or an expression (such as a + b or ).

Before diving into manipulating equations and inequalities, you will first look at the basic mathematical operations.

Linear Equations

PEMDAS
PEMDAS is a helpful acronym for remembering the proper order of operations. When working with an expression that contains only values (no variables), you must perform the operations in the order dictated by PEMDAS. The following question would require proficiency with PEMDAS:

Before answering this question, let’s review what each letter in PEMDAS stands for.

The acronym PEMDAS represents Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

When evaluating an expression, you must use the order as dictated by PEMDAS. However, note that there is no priority between multiplication and division, and no priority between addition and subtraction.

When choosing between these operations, always move from left to right.

The following list outlines what each operation represents:
- Parentheses ( ) or [ ]: parentheses are used to bracket off part of an expression from the rest of the expression. For example: (3 + 5) × 2.
- Exponents 4^3, 3^5, and so on: exponents tell you how many times you multiply the base by itself. 43 is the same as 4 × 4 × 4. 35 is the same as 3 × 3 × 3 × 3 × 3.
- Roots and so on: roots are the opposite of exponents. In terms of PEMDAS, roots take the same priority as exponents.

To get the square root of a number, you need to determine which value, when multiplied by itself, will yield the number under the square root.

- Multiplication 3 × 4; (3)(4); xy: Note that multiplication is the opposite of division. If you multiply 3 times 4 and then divide that result by 4, the result is 3.
- Division, : Next, you will evaluate any expressions that require multiplication or division. Note: There is no hierarchy between multiplication and division!

When presented with these two operations, always work from left to right:

- Addition/Subtraction 4 + 7 + 11; 8 – 3 = 5: Next, you will evaluate any expressions that require addition or subtraction. Note: Just as with multiplication and division, there is no priority between addition and subtraction! When presented with these two operations, always work from left to right:
7 − 9 + 5 → First, do 7 − 9 = −2, then do −2 + 5 = 3.

Example: Now that you’ve reviewed PEMDAS, let’s work through the question at the beginning of this section:

You see two expressions within parentheses, so first evaluate those expressions:

Next, you see a term raised to an exponent, so evaluate that term:

Now you are left with division and multiplication. Remember, there’s no priority between these two operations. Just work from left to right.

 

Simplifying Expressions
Before you dive into the process for solving for a variable, let’s review ways to simplify expressions.

An expression is some combination of variables, values, or both.

Examples of expressions are 3x + 7, , 9z3.
When presented with an expression, either by itself or as part of an equation, your first step should always be to simplify the expression.

Generally, there are three ways to do so:

Combine Like Terms
Simplify: 3x + 5z + 9x − 2z.

SOLUTION: Combine the xs and arrive at 12x. Combine the zs and arrive at 3z. The expression simplifies to 12x + 3z.
Simplify: 2(x+3) + 3(x + 3).

SOLUTION: Think of (x +3) as a variable, such as z. 2z + 3z = 5z. Therefore, 2(x + 3) + 3(x + 3) = 5(x + 3).

Find a Common Denominator

Simplify:

.

SOLUTION: Find a common denominator of 15 for both terms.

Simplify:

.

SOLUTION: Find a common denominator of xy for both terms.


Factor
To factor an expression, take out the factors common to all terms in the expression.

Example: Simplify: 6ab + 3a

SOLUTION: Each term has 3a as a factor. To see this, rewrite the expression as 3a × 2b + 3a × 1. Take 3a out of each term and arrive at 3a(2b + 1).
Simplify: 4x2 + 3x

SOLUTION: Each term shares a factor of x. Take x out of each term and arrive at x(4x + 3)

Basic Equations and Solving for a Variable
Whenever two expressions are set equal to each other, you have an equation:

The fundamental rule for all equations is the following: you can perform any operation on an equation as long as you perform that operation on both sides.

In the preceding example, if you divide both sides by 3, you will end up with 7 × 2 = 14. While the values on both sides of the equation changed, the equation itself is still true.

In algebra, equations will have variables, which are letters used to represent some unknown quantity in an equation.

When asked to solve for a variable, your goal will be isolate the variable by undoing the operations done to that variable.

To do so, you are going to use PEMDAS, but in reverse!

Let’s look at another example:
If , what is the value of x?

SOLUTION:
Step 1: Isolate by adding 3 to both sides of the equation.

Step 2: (x + 2) is within parentheses, so you should manipulate the equation to isolate the expression within the parentheses.

To do so, square both sides of the equation.


Systems of Equations: Combination and Substitution
Often, the GRE will present you with two or more equations with multiple variables and will ask you to solve for the value of one or more of the variables in those equations. This is called a system of equations. When working with a system of equations, your ultimate goal is to arrive at a situation similar to what you saw in the previous section: one equation with one variable.

To accomplish this, you can take two approaches: substitution or combination.

Substitution
Let’s say you are given the following question:

What is x?
Step 1: Express one variable in terms of the other variable.

Step 2: Take the expression for y and substitute it for y in the first equation: 3x + 2(9 − 2x) = 18.

Step 3: Solve for x.


Combination
Most systems of equations on the GRE can be solved with substitution, but you will sometimes need to use combination. Let’s look at an example:

Solve for a and z.
Step 1: When solving by combination, your ultimate goal is to arrive at the same coefficient for one of the variables. To do so, you will have to multiply each equation by a factor that will yield you a common coefficient for one of the variables.

In this case, let’s make the coefficients on z equal to 6:

Step 2: Multiply across one equation by negative 1.

 

Step 3: Add the equations to arrive at one equation and one variable (your ultimate goal!).

Substitute 0 for a into either equation to solve for x: