# Solving Equations

## Objectives

After completing this unit you should be able to:

• Solve equations by isolating variables through the use of addition, subtraction, multiplication, and division.
• Check that solutions are correct by the use of substitution.

## Introduction to Equations

Let's look back to the definition of algebra, a branch of mathematics that uses mathematical statements to describe relationships between things that vary with time and circumstances. The mathematical statements used are equations, which means that two expressions are equivalent. In order to understand the relationship being represented, you will need to be able to solve the equations used to describe these relationships.

What if we are given an equation such as x + 3 = 15, and asked to solve the equation? To solve an equation means to determine a numerical value for a variable (or variables, in equations that have more than one) that makes this statement true. This is the number that, when added to 3, gives the result of 15. With this simple equation you may be able to look at this and say the answer is 12, 12 + 3 = 15. However, not all equations are this simple.

In many introductory courses such as economics, chemistry, psychology, and others, you will be called upon to solve equations. For example, in economics you may need to determine amounts in production, consumption levels, material goods, scarce resource levels and specific economic units for individual industries, firms, or households.

The price, P, of a paperback book depends on the quantity, Q, purchased. The base price for purchasing one book is seven dollars. For each additional book purchased, the cost per book is reduced by 20 cents, or 0.20 dollars. The following algebraic equation represents the relationship between the price per book, P, of Q books purchased by:

P = (–.2)(Q – 1) + 7

In chemistry you may be required to determine the amount of product you should expect in a reaction given a certain amount of reactant. These kinds of problems can be set up as algebraic equations in which you will need to solve for an unknown variable. A sample problem here is:

For every nitrogen molecule that reacts with hydrogen, 2 ammonia molecules are produced. If N nitrogen molecules react with hydrogen, how many ammonia molecules, represented by the variable A, are formed? This can be represented by the following algebraic equation:

2 N = A

Both of these examples contain more than one variable and represent the types of equations you use in your course work. In this unit, we will begin our discussion solving equations that contain single variables. We saw that the single variable equation x + 3 = 15 could be solved by inspection, but many single variable equations are far more complex than this. For these more complex equations, we will need to use systematic methods for finding solutions:

 Steps for Solving Equations Combine like terms. Isolate the terms that contain the variable you wish to solve for. Isolate the variable you wish to solve for. Substitute your answer into the original equation and check that it works.

In the sections associated with this unit you can:

When you are ready, try the practice for this unit.