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Study Guides > Mathematics for the Liberal Arts Corequisite

Solving Equations With Integers Using Properties of Equality

Learning Outcomes

  • Solve equations using the addition and subtraction properties of equality
  • Model the division property of equality
  • Solve equations using the multiplication and division properties of equality

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.

x+4=12x+4=12                                  y5=9y--5=9

x+44=124x+4\color{red}{--4}=12\color{red}{--4}                    y5+5=9+5y--5\color{red}{+5}=9\color{red}{+5}

x=8x=8                                            y=14y=14

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
For any numbers a,b,c\text{For any numbers }a,b,c, if a=b then ac=bc\text{if }a=b\text{ then }a-c=b-c. For any numbers a,b,c\text{For any numbers }a,b,c, if a=b then a+c=b+c\text{if }a=b\text{ then }a+c=b+c.
 

example

Solve: y+9=5y+9=5 Solution
y+9=5y+9=5
Subtract 99 from each side to undo the addition. y+99=59y+9\color{red}{--9}=5\color{red}{--9}
Simplify. y=4y=--4
Check the result by substituting 4-4 into the original equation.
y+9=5y+9=5
Substitute 4−4 for y 4+9=?5-4+9\stackrel{?}{=}5
5=55=5\quad\checkmark
Since y=4y=-4 makes y+9=5y+9=5 a true statement, we found the solution to this equation.  
 

try it

[ohm_question]141721[/ohm_question]  
 

example

Solve: a6=8a - 6=-8

Answer: Solution

a6=8a--6=--8
Add 66 to each side to undo the subtraction. a6+6=8+6a--6\color{red}{+6}=--8\color{red}{+6}
Simplify. a=2a=--2
Check the result by substituting 2-2 into the original equation: a6=8a--6=--8
Substitute 2-2 for aa 26=?8--2--6\stackrel{?}{=}--8
8=8--8=--8\quad\checkmark
The solution to a6=8a - 6=-8 is 2-2. Since a=2a=-2 makes a6=8a - 6=-8 a true statement, we found the solution to this equation.

 

try it

[ohm_question]146557[/ohm_question]
In the following video we show more examples of how to solve linear equations involving integers using the addition and subtraction properties of equality. https://youtu.be/xGfOlCluPDo

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 xa=bx-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. This image has two columns. In the first column are two identical envelopes. In the second column there are six blue circles, randomly placed. 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=36\div2=3). What equation models the situation shown in the figure below? 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=62x=6. This image has two columns. In the first column are two identical envelopes. In the second column there are six blue circles, randomly placed. Under the figure is two times x equals 6. We can divide both sides of the equation by 22 as we did with the envelopes and counters.

2x2=62\Large{\frac{2x}{\color{red}{2}}=\frac{6}{\color{red}{2}}}

x=3x=3

We found that each envelope contains 3 counters.\text{3 counters.} Does this check? We know 23=62\cdot 3=6, so it works. Three counters in each of two envelopes does equal six. Another example is shown below. This image has two columns. In the first column are three envelopes. In the second column there are four rows of three blue circles. Underneath the image is the equation 3x equals 12. Now we have 33 identical envelopes and 12 counters.\text{12 counters.} How many counters are in each envelope? We have to separate the 12 counters\text{12 counters} into 3 groups.\text{3 groups.} Since 12÷3=412\div 3=4, there must be 4 counters\text{4 counters} in each envelope. This image has two columns. In the first column are four envelopes. In the second column there are twelve blue circles. The equation that models the situation is 3x=123x=12. We can divide both sides of the equation by 33.

3x3=123\Large{\frac{3x}{\color{red}{3}}=\frac{12}{\color{red}{3}}}

x=4x=4

Does this check? It does because 34=123\cdot 4=12.

example

Write an equation modeled by the envelopes and counters, and then solve it. This image has two columns. In the first column are four envelopes. In the second column there are 8 blue circles. Solution There are 4 envelopes,\text{4 envelopes,} or 44 unknown values, on the left that match the 8 counters\text{8 counters} on the right. Let’s call the unknown quantity in the envelopes xx.
Write the equation. 4x=84x=8
Divide both sides by 44. 4x4=84\Large{\frac{4x}{\color{red}{4}}=\frac{8}{\color{red}{4}}}
Simplify. x=2x=2
There are 2 counters\text{2 counters} in each envelope.
 

try it

Write the equation modeled by the envelopes and counters. Then solve it. This image has two columns. In the first column are four envelopes. In the second column there are 12 blue circles. 4x=12[/latex];[latex]x=34x=12[/latex]; [latex]x=3 Write the equation modeled by the envelopes and counters. Then solve it. This image has two columns. In the first column are three envelopes. In the second column there are six blue circles. 3x=6[/latex];[latex]x=23x=6[/latex]; [latex]x=2

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

For any numbersa,b,c,andc0,Ifa=b thenac=bc.\begin{array}{ccc}\text{For any numbers}& a,b,c,\text{and}& c\ne 0,\\ \hfill \text{If}& a=b\text{ then}& \large{\frac{a}{c}=\frac{b}{c}}.\end{array}
 

example

Solve: 7x=49\text{Solve: }7x=-49.

Answer: Solution To isolate xx, we need to undo multiplication.

7x=497x=--49
Divide each side by 77. 7x7=497\Large{\frac{7x}{\color{red}{7}}=\frac{--49}{\color{red}{7}}}
Simplify. x=7x=--7
Check the solution.
7x=497x=-49
Substitute 7−7 for x. 7(7)=?497\left(-7\right)\stackrel{?}{=}-49
49=49-49=-49\quad\checkmark
Therefore, 7-7 is the solution to the equation.

 

try it

[ohm_question]146560[/ohm_question]
 

example

Solve: 3y=63-3y=63.

Answer: Solution To isolate yy, we need to undo the multiplication.

3y=63--3y=63
Divide each side by 3−3. 3y3=633\Large{\frac{--3y}{\color{red}{--3}}=\frac{63}{\color{red}{--3}}}
Simplify y=21y=--21
Check the solution.
3y=63-3y=63
Substitute 21−21 for y. 3(21)=?63-3\left(-21\right)\stackrel{?}{=}63
63=6363=63\quad\checkmark
Since this is a true statement, y=21y=-21 is the solution to the equation.

 

try it

[ohm_question]146561[/ohm_question]
Watch the following video to see more examples of how to use the division and multiplication properties to solve equations with integers. https://youtu.be/rZuvbYO3sV8

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