Learning Objectives
- Determine whether an integer is a solution of an equation
- Solve equations with integers using the Addition and Subtraction Properties of Equality
- Model the Division Property of Equality
- Solve equations using the Division Property of Equality
- Translate to an equation and solve
Before you get started, take this readiness quiz.
If you missed this problem, review Example 3.22.
If you missed this problem, review Example 2.33.- Translate into an algebraic expression less than
If you missed this problem, review Table 1.3.
Determine Whether a Number is a Solution of an Equation
In Solve Equations with the Subtraction and Addition Properties of Equality, we saw that a solution of an equation is a value of a variable that makes a true statement when substituted into that equation. In that section, we found solutions that were whole numbers. Now that we’ve worked with integers, we’ll find integer solutions to equations.
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 or an integer.
How To
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 it is true, the number is a solution.
- If it is not true, the number is not a solution.
Example 3.60
Determine whether each of the following is a solution of
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Determine whether each of the following is a solution of
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Determine whether each of the following is a solution of
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Solve Equations with Integers Using the Addition and Subtraction Properties of Equality
In Solve Equations with the Subtraction and Addition Properties of Equality, we solved equations similar to the two shown here using the Subtraction and Addition Properties of Equality. Now we can use them again with integers.
When you add or subtract the same quantity from both sides of an equation, you still have equality.
Properties of Equalities
Subtraction Property of Equality | Addition Property of Equality |
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Example 3.61
Solve:
Solve:
Solve:
Example 3.62
Solve:
Solve:
Solve:
Model the Division Property of Equality
All of the equations we have solved so far have been of the form or We were able to isolate the variable by adding or subtracting the constant term. Now we’ll see how to solve equations that involve division.
We will model an equation with envelopes and counters in Figure 3.21.
Here, there are two identical envelopes that contain the same number of counters. Remember, the left side of the workspace must equal the right side, but the counters on the left side are “hidden” in the envelopes. So how many counters are in each envelope?
To determine the number, separate the counters on the right side into groups of the same size. So counters divided into groups means there must be counters in each group (since
What equation models the situation shown in Figure 3.22? There are two envelopes, and each contains counters. Together, the two envelopes must contain a total of counters. So the equation that models the situation is
We can divide both sides of the equation by as we did with the envelopes and counters.
We found that each envelope contains Does this check? We know so it works. Three counters in each of two envelopes does equal six.
Figure 3.23 shows another example.
Now we have identical envelopes and How many counters are in each envelope? We have to separate the into Since there must be in each envelope. See Figure 3.24.
The equation that models the situation is We can divide both sides of the equation by
Does this check? It does because
Manipulative Mathematics
Example 3.63
Write an equation modeled by the envelopes and counters, and then solve it.
Write the equation modeled by the envelopes and counters. Then solve it.
Write the equation modeled by the envelopes and counters. Then solve it.
Solve Equations Using the Division Property of Equality
The previous examples lead to the Division Property of Equality. When you divide both sides of an equation by any nonzero number, you still have equality.
Division Property of Equality
Example 3.64
Solve:
Solve:
Example 3.65
Solve:
Solve:
Solve:
Translate to an Equation and Solve
In the past several examples, we were given an equation containing a variable. In the next few examples, we’ll have to first translate word sentences into equations with variables and then we will solve the equations.
Example 3.66
Translate and solve: five more than is equal to
Translate and solve:
Seven more than is equal to .
Translate and solve:
Example 3.67
Translate and solve: the difference of and is
Translate and solve:
The difference of and is .
Translate and solve:
The difference of and is .
Example 3.68
Translate and solve: the number is the product of and
Translate and solve:
The number is the product of and .
Translate and solve:
The number is the product of and .
Media Access Additional Online Resources
Section 3.5 Exercises
Practice Makes Perfect
Determine Whether a Number is a Solution of an Equation
In the following exercises, determine whether each number is a solution of the given equation.
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Solve Equations Using the Addition and Subtraction Properties of Equality
In the following exercises, solve for the unknown.
Model the Division Property of Equality
In the following exercises, write the equation modeled by the envelopes and counters and then solve it.
Solve Equations Using the Division Property of Equality
In the following exercises, solve each equation using the division property of equality and check the solution.
Translate to an Equation and Solve
In the following exercises, translate and solve.
Nine more than is equal to 5.
The sum of two and is .
The difference of and is .
The number −54 is the product of −9 and .
The product of and −18 is 36.
−2 plus is equal to 1.
Thirteen less than is .
Mixed Practice
In the following exercises, solve.
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Everyday Math
Cookie packaging A package of has equal rows of cookies. Find the number of cookies in each row, by solving the equation
Kindergarten class Connie’s kindergarten class has She wants them to get into equal groups. Find the number of children in each group, by solving the equation
Writing Exercises
Is modeling the Division Property of Equality with envelopes and counters helpful to understanding how to solve the equation Explain why or why not.
Suppose you are using envelopes and counters to model solving the equations and Explain how you would solve each equation.
Frida started to solve the equation by adding to both sides. Explain why Frida’s method will not solve the equation.
Raoul started to solve the equation by subtracting from both sides. Explain why Raoul’s method will not solve the equation.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?