# Solving Linear Equations

## Use a Problem Solving Strategy

Have you ever had any negative experiences in the past with word problems? When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. Realize that your negative experiences with word problems are in your past. To move forward you need to calm your fears and change your negative feelings.
Start with a fresh slate and begin to think positive thoughts. Repeating some of the following statements may be helpful to turn your thoughts positive. Thinking positive thoughts is a first step towards success.
  I think I can! I think I can!
  While word problems were hard in the past, I think I can try them now.
  I am better prepared now—I think I will begin to understand word problems.
  I am able to solve equations because I practiced many problems and I got help when I needed it—I can try that  with word problems.
  It may take time, but I can begin to solve word problems.
You are now well prepared and you are ready to succeed. If you take control and believe you can be successful, you will be able to master word problems.

### Use a Problem Solving Strategy for Word Problems

Now that we can solve equations, we are ready to apply our new skills to word problems. We will develop a strategy we can use to solve any word problem successfully.
We summarize an effective strategy for problem solving.

### Solve Number Word Problems

We will now apply the problem solving strategy to “number word problems.” Number word problems give some clues about one or more numbers and we use these clues to write an equation. Number word problems provide good practice for using the Problem Solving Strategy.
Some number word problems ask us to find two or more numbers. It may be tempting to name them all with different variables, but so far, we have only solved equations with one variable. In order to avoid using more than one variable, we will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.
Some number problems involve consecutive integers. Consecutive integers are integers that immediately follow each other. Examples of consecutive integers are:
Notice that each number is one more than the number preceding it. Therefore, if we define the first integer as n, the next consecutive integer is  The one after that is one more than  so it is  which is 
We will use this notation to represent consecutive integers in the next example.
Now that we have worked with consecutive integers, we will expand our work to include consecutive even integers and consecutive odd integers. Consecutive even integers are even integers that immediately follow one another. Examples of consecutive even integers are:
Notice each integer is two more than the number preceding it. If we call the first one n, then the next one is  The one after that would be  or 
Consecutive odd integers are odd integers that immediately follow one another. Consider the consecutive odd integers 63, 65, and 67.
Does it seem strange to have to add two (an even number) to get the next odd number? Do we get an odd number or an even number when we add 2 to 3? to 11? to 47?
Whether the problem asks for consecutive even numbers or odd numbers, you do not have to do anything different. The pattern is still the same—to get to the next odd or the next even integer, add two.
When a number problem is in a real life context, we still use the same strategies that we used for the previous examples.

### Solve Percent Applications

There are several methods to solve percent equations. In algebra, it is easiest if we just translate English sentences into algebraic equations and then solve the equations. Be sure to change the given percent to a decimal before you use it in the equation.
Now that we have a problem solving strategy to refer to, and have practiced solving basic percent equations, we are ready to solve percent applications. Be sure to ask yourself if your final answer makes sense—since many of the applications we will solve involve everyday situations, you can rely on your own experience.
Remember to put the answer in the form requested. In the next example we are looking for the percent.
It is often important in many fields—business, sciences, pop culture—to talk about how much an amount has increased or decreased over a certain period of time. This increase or decrease is generally expressed as a percent and called the percent change.
To find the percent change, first we find the amount of change, by finding the difference of the new amount and the original amount. Then we find what percent the amount of change is of the original amount.
Applications of discount and mark-up are very common in retail settings.
When you buy an item on sale, the original price has been discounted by some dollar amount. The discount rate, usually given as a percent, is used to determine the amount of the discount. To determine the amount of discount, we multiply the discount rate by the original price.
The price a retailer pays for an item is called the original cost. The retailer then adds a mark-up to the original cost to get the list price, the price he sells the item for. The mark-up is usually calculated as a percent of the original cost. To determine the amount of mark-up, multiply the mark-up rate by the original cost.

### Solve Simple Interest Applications

Interest is a part of our daily lives. From the interest earned on our savings to the interest we pay on a car loan or credit card debt, we all have some experience with interest in our lives.
The amount of money you initially deposit into a bank is called the principal, P, and the bank pays you interest, I. When you take out a loan, you pay interest on the amount you borrow, also called the principal.
In either case, the interest is computed as a certain percent of the principal, called the rate of interest, r. The rate of interest is usually expressed as a percent per year, and is calculated by using the decimal equivalent of the percent. The variable t, (for time) represents the number of years the money is saved or borrowed.
Interest is calculated as simple interest or compound interest. Here we will use simple interest.
The formula we use to calculate interest is  To use the formula we substitute in the values for variables that are given, and then solve for the unknown variable. It may be helpful to organize the information in a chart.
There may be times when we know the amount of interest earned on a given principal over a certain length of time, but we do not know the rate.
In the next example, we are asked to find the principal—the amount borrowed.

### Key Concepts

1. How To Use a Problem Solving Strategy for Word Problems


2. How To Find Percent Change


3. Discount
4. Mark-up
5. Simple Interest
If an amount of money, P, called the principal, is invested or borrowed for a period of t years at an annual interest rate r, the amount of interest, I, earned or paid is:


### Practice Makes Perfect

Use a Problem Solving Strategy for Word Problems
In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question.
Solve Number Word Problems
In the following exercises, solve each number word problem.
Solve Percent Applications
In the following exercises, translate and solve.
In the following exercises, solve.
In the following exercises, solve.
In the following exercises, find ⓐ the amount of discount and ⓑ the sale price.
In the following exercises, find ⓐ the amount of discount and ⓑ the discount rate (Round to the nearest tenth of a percent if needed.)
In the following exercises, find ⓐ the amount of the mark-up and ⓑ the list price.
Solve Simple Interest Applications
In the following exercises, solve.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objective of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?