By the end of this section, you will be able to:
We are now ready to “get to the good stuff.” You have the basics down and are ready to begin one of the most important topics in algebra: solving equations. The applications are limitless and extend to all careers and fields. Also, the skills and techniques you learn here will help improve your critical thinking and problem-solving skills. This is a great benefit of studying mathematics and will be useful in your life in ways you may not see right now.
We began our work solving equations in previous chapters. It has been a while since we have seen an equation, so we will review some of the key concepts before we go any further.
We said that solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle.
A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.
In the earlier sections, we listed the steps to determine if a value is a solution. We restate them here.
Step 1. Substitute the number for the variable in the equation.
Step 2. Simplify the expressions on both sides of the equation.
Step 3. Determine whether the resulting equation is true.
If it is true, the number is a solution.
If it is not true, the number is not a solution.
We introduced the Subtraction and Addition Properties of Equality in Solving Equations Using the Subtraction and Addition Properties of Equality. In that section, we modeled how these properties work and then applied them to solving equations with whole numbers. We used these properties again each time we introduced a new system of numbers. Let’s review those properties here.
Subtraction Property of Equality
For all real numbers a,b, and c, if a=b, then a−c=b−c.
Addition Property of Equality
For all real numbers a,b, and c, if a=b, then a+c=b+c.
When you add or subtract the same quantity from both sides of an equation, you still have equality.
We introduced the Subtraction Property of Equality earlier by modeling equations with envelopes and counters. Figure 8.2 models the equation x+3=8.
The goal is to isolate the variable on one side of the equation. So we ‘took away’ 3 from both sides of the equation and found the solution x=5.
Some people picture a balance scale, as in Figure 8.3, when they solve equations.
The quantities on both sides of the equal sign in an equation are equal, or balanced. Just as with the balance scale, whatever you do to one side of the equation you must also do to the other to keep it balanced.
Let’s review how to use Subtraction and Addition Properties of Equality to solve equations. We need to isolate the variable on one side of the equation. And we check our solutions by substituting the value into the equation to make sure we have a true statement.
In the original equation in the previous example, 11 was added to the x, so we subtracted 11 to ‘undo’ the addition. In the next example, we will need to ‘undo’ subtraction by using the Addition Property of Equality.
Now let’s review solving equations with fractions.
In Solve Equations with Decimals, we solved equations that contained decimals. We’ll review this next.
In the examples up to this point, we have been able to isolate the variable with just one operation. Many of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation before using the Subtraction or Addition Properties of Equality. You should always simplify as much as possible before trying to isolate the variable.
In previous chapters, we translated word sentences into equations. The first step is to look for the word (or words) that translate(s) to the equal sign. Table 8.1 reminds us of some of the words that translate to the equal sign.
Let’s review the steps we used to translate a sentence into an equation.
Step 1. Locate the “equals” word(s). Translate to an equal sign.
Step 2. Translate the words to the left of the “equals” word(s) into an algebraic expression.
Step 3. Translate the words to the right of the “equals” word(s) into an algebraic expression.
Now we are ready to try an example.
In most of the application problems we solved earlier, we were able to find the quantity we were looking for by simplifying an algebraic expression. Now we will be using equations to solve application problems. We’ll start by restating the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve. When assigning a variable, choose a letter that reminds you of what you are looking for.
Step 1. Read the problem. Make sure you understand all the words and ideas.
Step 2. Identify what you are looking for.
Step 3. Name what you are looking for. Choose a variable to represent that quantity.
Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
Step 5. Solve the equation using good algebra techniques.
Step 6. Check the answer in the problem and make sure it makes sense.
Step 7. Answer the question with a complete sentence.
The Links to Literacy activity, “The 100-pound Problem”, will provide you with another view of the topics covered in this section.
Solving One Step Equations By Addition and Subtraction
Solve One Step Equations By Add and Subtract Whole Numbers (Variable on Left)
Solve One Step Equations By Add and Subtract Whole Numbers (Variable on Right)
Solve Equations Using the Subtraction and Addition Properties of Equality
In the following exercises, determine whether the given value is a solution to the equation.
In the following exercises, solve each equation.
Solve Equations that Need to be Simplified
In the following exercises, solve each equation.
Translate to an Equation and Solve
In the following exercises, translate to an equation and then solve.
Translate and Solve Applications
In the following exercises, translate into an equation and solve.
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.