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# Simplifying Expressions and Solving Equation Word Problems Study Guide

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## Introduction to Simplifying Expressions and Solving Equation Word Problems

Mathematics may be defined as the economy of counting. There is no problem in the whole of mathematics which cannot be solved by direct counting.

—ERNST MACH (1838–1916)

This lesson reviews the key words and phrases for the basic operations and provides examples and tips on simplifying algebraic expressions and the equation solving steps. Equation word problems are modeled with explanations to help your understanding in this type of question.

### Key Words and Phrases

Translating expressions from words into mathematical symbols was covered in Lesson 1. The following chart below summarizes the key words and phrases studied in that lesson for the four basic operations and the equal sign.

Refer to this chart when you are changing sentences in words to math equations.

#### Tip:

In algebra, the number in front of the letter is called the coefficient and the letter is called the variable. In the expression 8x, 8 is the coefficient and x is the variable.

## Simplifying Expressions - Combining Like Terms and The Distributive Property

Two important processes to know when you are simplifying expressions are combining like terms and the distributive property.

### Combining Like Terms

Terms, in mathematics, are numbers and symbols that are separated by addition and subtraction.

The expressions 3, 5x, and 7xy are each one term.
The expressions 2x + 3, and x – 7 each have two terms.
The expression 7x + 5y – 9 has three terms.

Like terms are terms with the same variable and exponent. Like terms can be combined by addition and subtraction. To do this, add or subtract the coefficients and keep the variable the same. For example 3x + 5x = 8x, and 6y2 – 4y2 = 2y2.

#### Tip:

Be sure to combine only like terms. Terms without the exact same variable and exponent cannot be combined: 5x2 and 6x cannot be combined because the exponents are not the same.

### Distributive Property

The distributive property is used when a value needs to be multiplied, or distributed, to more than one term. For example, in the expression 3(x + 10), the number 3 needs to be multiplied by the term x and the term 10. The use of arrows can help in this process, as shown in the following figure.

The result becomes 3 × x + 3 × 10, which simplifies to 3x + 30.

### Solving Equations

When you are solving equations, the goal is to get the letter, or variable, by itself. This is called isolating the variable. Each of the following examples goes through the process of isolating the variable for different types of equations.

#### Tip:

One of the most important rules in equation solving is to do the same thing on both sides of the equation. For example, if you divide on one side to get the variable alone, divide the other side by the same number. This keeps the equation balanced and will lead to the correct solution.

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