5.5: Solve Equations with Decimals
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- May 28, 2023
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By the end of this section, you will be able to:
- Determine whether a decimal is a solution of an equation
- Solve equations with decimals
- Translate to an equation and solve
Be Prepared 5.10
Before you get started, take this readiness quiz.
Evaluate
If you missed this problem, review Example 4.77.
Be Prepared 5.11
Evaluate
If you missed this problem, review Example 3.41.
Be Prepared 5.12
Solve
If you missed this problem, review Example 4.99.
Determine Whether a Decimal is a Solution of an Equation
Solving equations with decimals is important in our everyday lives because money is usually written with decimals. When applications involve money, such as shopping for yourself, making your family’s budget, or planning for the future of your business, you’ll be solving equations with decimals.
Now that we’ve worked with decimals, we are ready to find solutions to equations involving decimals. The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, a fraction, or a decimal. We’ll list these steps here again for easy reference.
How To
Determine whether a number is a solution to an equation.
- 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 so, the number is a solution.
- If not, the number is not a solution.
Example 5.40
Determine whether each of the following is a solution of
ⓐ
- Answer
ⓐ Subtract. Since
does not result in a true equation, is not a solution to the equation. ⓑ Subtract.
Since
| ⓒ | |
| Subtract. |
Since
Try It 5.79
Determine whether each value is a solution of the given equation.
Try It 5.80
Determine whether each value is a solution of the given equation.
Solve Equations with Decimals
In previous chapters, we solved equations using the Properties of Equality. We will use these same properties to solve equations with decimals.
Properties of Equality
| Subtraction Property of Equality For any numbers If |
Addition Property of Equality For any numbers If |
| The Division Property of Equality For any numbers If |
The Multiplication Property of Equality For any numbers If |
When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.
Example 5.41
Solve:
- Answer
We will use the Subtraction Property of Equality to isolate the variable.
Simplify. Check: Simplify. Since
makes a true statement, we know we have found a solution to this equation.
Try It 5.81
Solve:
Try It 5.82
Solve:
Example 5.42
Solve:
- Answer
We will use the Addition Property of Equality.
Add 4.75 to each side, to undo the subtraction. Simplify. Check: Since the result is a true statement,
is a solution to the equation.
Try It 5.83
Solve:
Try It 5.84
Solve:
Example 5.43
Solve:
- Answer
We will use the Division Property of Equality.
Use the Properties of Equality to find a value for
We must divide both sides by 0.8 to isolate n. Simplify. Check: Since
makes a true statement, we know we have a solution.
Try It 5.85
Solve:
Try It 5.86
Solve:
Example 5.44
Solve:
- Answer
We will use the Multiplication Property of Equality.
Here, p is divided by −1.8. We must multiply by −1.8 to isolate p Multiply. Check: A solution to
is
Try It 5.87
Solve:
Try It 5.88
Solve:
Translate to an Equation and Solve
Now that we have solved equations with decimals, we are ready to translate word sentences to equations and solve. Remember to look for words and phrases that indicate the operations to use.
Example 5.45
Translate and solve: The difference of
- Answer
Translate. Add to both sides of the equation. Simplify. Check: Is the difference of and 4.3 equal to 2.1? Let Is the difference of 6.4 and 4.3 equal to 2.1? Translate. Simplify.
Try It 5.89
Translate and solve: The difference of
Try It 5.90
Translate and solve: The difference of
Example 5.46
Translate and solve: The product of
- Answer
Translate. Divide both sides by . Simplify. Check: Is the product of −3.1 and equal to ? Let Is the product of and equal to ? Translate. Simplify.
Try It 5.91
Translate and solve: The product of
Try It 5.92
Translate and solve: The product of
Example 5.47
Translate and solve: The quotient of
- Answer
Translate. Multiply both sides by . Simplify. Check: Is the quotient of and equal to ? Let Is the quotient of and equal to ? Translate. Simplify.
Try It 5.93
Translate and solve: The quotient of
Try It 5.94
Translate and solve: The quotient of
Example 5.48
Translate and solve: The sum of
- Answer
Translate. Subtract from each side. Simplify. Check: Is the sum and equal to ? Let Is the sum and equal to ? Translate. Simplify.
Try It 5.95
Translate and solve: The sum of
Try It 5.96
Translate and solve: The sum of
Media
Section 5.4 Exercises
Practice Makes Perfect
Determine Whether a Decimal is a Solution of an Equation
In the following exercises, determine whether each number is a solution of the given equation.
ⓐ
ⓐ
ⓐ
ⓐ
Solve Equations with Decimals
In the following exercises, solve the equation.
Mixed Practice
In the following exercises, solve the equation. Then check your solution.
Translate to an Equation and Solve
In the following exercises, translate and solve.
The difference of
The difference
The product of
The product of
The quotient of
The quotient of
The sum of
The sum of
Everyday Math
Shawn bought a pair of shoes on sale for
Mary bought a new refrigerator. The total price including sales tax was
Writing Exercises
Think about solving the equation
Think about solving the equation
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?
