3.5 Solve Equations Using Integers; The Division Property of Equality
Learning Objectives
By the end of this section, you will be able to:
- 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
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.
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.
Model the Division Property of Equality
All of the equations we have solved so far have been of the form x+a=bx+a=b or x−a=b.x−a=b. 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.
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 22 groups of the same size. So 66 counters divided into 22 groups means there must be 33 counters in each group (since 6÷2=3).6÷2=3).
What equation models the situation shown in Figure 3.22? There are two envelopes, and each contains xx counters. Together, the two envelopes must contain a total of 66 counters. So the equation that models the situation is 2x=6
Figure 3.22
We can divide both sides of the equation by 22 as we did with the envelopes and counters.
We found that each envelope contains 3 counters.3 counters. Does this check? We know 2⋅3=6,2·3=6, so it works. Three counters in each of two envelopes does equal six.
Figure 3.23 shows another example.
Figure 3.23
Now we have 33 identical envelopes and 12 counters.12 counters. How many counters are in each envelope? We have to separate the 12 counters12 counters into 3 groups.3 groups. Since 12÷3=4,12÷3=4, there must be 4 counters4 counters in each envelope. See Figure 3.24.
Figure 3.24
The equation that models the situation is 3x=12.3x=12. We can divide both sides of the equation by 3.3.
Does this check? It does because 3⋅4=12.
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.
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.
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.
- 4x−2=6
ⓐx=−2
ⓑx=−1
ⓒx=2 - 4y−10=−14
ⓐy=−6
ⓑy=−1
ⓒy=1 - 9a+27=−63
ⓐa=6
ⓑa=−6
ⓒa=−10 - 7c+42=−56
ⓐ c=2
ⓑ c=−2
ⓒ c=−14
Solve Equations Using the Addition and Subtraction Properties of Equality
In the following exercises, solve for the unknown.
- n+12=5
- m+16=2
- p+9=−8
- q+5=−6
- u−3=−7
- v−7=−8
- h−10=−4
- k−9=−5
- x+(−2)=−18
- y+(−3)=−10
- r−(−5)=−9
- s−(−2)=−11
Model the Division Property of Equality
In the following exercises, write the equation modeled by the envelopes and counters and then solve it.
301.
302.
303.
304.
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.
- 5x=45
- 4p=64
- −7c=56
- −9x=54
- −14p=−42
- −8m=−40
- −120=10q
- −75=15y
- 24x=480
- 18n=540
- −3z=0
- 4u=0
Translate to an Equation and Solve
In the following exercises, translate and solve.
- Four more than n is equal to 1.
- Nine more than m is equal to 5.
- The sum of eight and p is −3.
- The sum of two and q is −7.
- The difference of a and three is −14.
- The difference of b and 5 is −2.
- The number −42 is the product of −7 and x.
- The number −54 is the product of −9 and y.
- The product of -15 and f is 75.
- The product of −18 and g is 36.
- −6 plus c is equal to 4.
- −2 plus d is equal to 1.
- Nine less than m is −4.
- Thirteen less than n is −10.
Mixed Practice
In the following exercises, solve.
331.
ⓐx+2=10
ⓑ2x=10
332.
ⓐy+6=12
ⓑ6y=12
333.
ⓐ−3p=27
ⓑp−3=27
334.
ⓐ−2q=34
ⓑq−2=34
- a−4=16
- b−1=11
- −8m=−56
- −6n=−48
- −39=u+13
- −100=v+25
- 11r=−99
- 15s=−300
- 100=20d
- 250=25n
- −49=x−7
- 64=y−4
Everyday Math
- Cookie packaging A package of 51 cookies has 3 equal rows of cookies. Find the number of cookies in each row, c, by solving the equation 3c=51.
- Kindergarten class Connie’s kindergarten class has 24 children. She wants them to get into 4 equal groups. Find the number of children in each group, g, by solving the equation 4g=24.
Writing Exercises
- Is modeling the Division Property of Equality with envelopes and counters helpful to understanding how to solve the equation 3x=15? Explain why or why not.
- Suppose you are using envelopes and counters to model solving the equations x+4=12 and 4x=12. Explain how you would solve each equation.
- Frida started to solve the equation −3x=36 by adding 3 to both sides. Explain why Frida’s method will not solve the equation.
- Raoul started to solve the equation 4y=40 by subtracting 4 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?