1.7.b – Solving Equations With Integers Using Properties of Equality

Learning Outcomes

Solve equations with integers using the addition and subtraction properties of equality
Model the division property of equality
Solve equations using the multiplication and division properties of equality
Translate word problems to equations and solve

Solve Equations with Integers Using the Addition and Subtraction Properties of Equality

We solved equations similar to the two shown here, where only whole numbers were used, but now we can also solve equations using integers.

[latex]x+4=12[/latex] [latex]y--5=9[/latex]

[latex]x+4\color{red}{--4}=12\color{red}{--4}[/latex] [latex]y--5\color{red}{+5}=9\color{red}{+5}[/latex]

[latex]x=8[/latex] [latex]y=14[/latex]

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
[latex]\text{For any numbers }a,b,c[/latex], [latex]\text{if }a=b\text{ then }a-c=b-c[/latex].	[latex]\text{For any numbers }a,b,c[/latex], [latex]\text{if }a=b\text{ then }a+c=b+c[/latex].

Subtraction Property of Equality

Addition Property of Equality

[latex]\text{For any numbers }a,b,c[/latex],

[latex]\text{if }a=b\text{ then }a-c=b-c[/latex].

[latex]\text{For any numbers }a,b,c[/latex],

[latex]\text{if }a=b\text{ then }a+c=b+c[/latex].

example

Solve: [latex]y+9=5[/latex]

Solution

	[latex]y+9=5[/latex]
Subtract [latex]9[/latex] from each side to undo the addition.	[latex]y+9\color{red}{--9}=5\color{red}{--9}[/latex]
Simplify.	[latex]y=--4[/latex]

Check the result by substituting [latex]-4[/latex] into the original equation.

	[latex]y+9=5[/latex]
Substitute [latex]−4[/latex] for y	[latex]-4+9\stackrel{?}{=}5[/latex]
	[latex]5=5\quad\checkmark[/latex]

Since [latex]y=-4[/latex] makes [latex]y+9=5[/latex] a true statement, we found the solution to this equation.

try it

example

Solve: [latex]a - 6=-8[/latex]

Show Solution

try it

In the following video we show more examples of how to solve linear equations involving integers using the addition and subtraction properties of equality.

Model the Division Property of Equality

All of the equations we have solved so far have been of the form [latex]x+a=b[/latex] or [latex]x-a=b[/latex]. 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 [latex]2[/latex] groups of the same size. So [latex]6[/latex] counters divided into [latex]2[/latex] groups means there must be [latex]3[/latex] counters in each group (since [latex]6\div2=3[/latex]).

What equation models the situation shown in the figure below? There are two envelopes, and each contains [latex]x[/latex] counters. Together, the two envelopes must contain a total of [latex]6[/latex] counters. So the equation that models the situation is [latex]2x=6[/latex].

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 [latex]2[/latex] as we did with the envelopes and counters.

[latex]\Large{\frac{2x}{\color{red}{2}}=\frac{6}{\color{red}{2}}}[/latex]

[latex]x=3[/latex]

We found that each envelope contains [latex]\text{3 counters.}[/latex] Does this check? We know [latex]2\cdot 3=6[/latex], 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 [latex]3[/latex] identical envelopes and [latex]\text{12 counters.}[/latex] How many counters are in each envelope? We have to separate the [latex]\text{12 counters}[/latex] into [latex]\text{3 groups.}[/latex] Since [latex]12\div 3=4[/latex], there must be [latex]\text{4 counters}[/latex] 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 [latex]3x=12[/latex]. We can divide both sides of the equation by [latex]3[/latex].

[latex]\Large{\frac{3x}{\color{red}{3}}=\frac{12}{\color{red}{3}}}[/latex]

[latex]x=4[/latex]

Does this check? It does because [latex]3\cdot 4=12[/latex].

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 [latex]\text{4 envelopes,}[/latex] or [latex]4[/latex] unknown values, on the left that match the [latex]\text{8 counters}[/latex] on the right. Let’s call the unknown quantity in the envelopes [latex]x[/latex].

Write the equation.	[latex]4x=8[/latex]
Divide both sides by [latex]4[/latex].	[latex]\Large{\frac{4x}{\color{red}{4}}=\frac{8}{\color{red}{4}}}[/latex]
Simplify.	[latex]x=2[/latex]

There are [latex]\text{2 counters}[/latex] 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.

[latex]4x=12[/latex]; [latex]x=3[/latex]

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.

[latex]3x=6[/latex]; [latex]x=2[/latex]

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

[latex]\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}[/latex]

example

[latex]\text{Solve: }7x=-49[/latex].

Show Solution

try it

example

Solve: [latex]-3y=63[/latex].

Show Solution

try it

Watch the following video to see more examples of how to use the division and multiplication properties to solve equations with integers.

Translating and Solving Word Problems

Now let’s apply these skills to solving word problems involving integers and basic algebraic equations.

Example

Jessa collects small potted plants. She loves both succulents and tropical plants. The number of succulents is six more than five times the number of tropical plants. Let [latex]t[/latex] represent the number of tropical plants, and write an expression for the number of succulents. Then solve for the number of tropical plants in Jessa’s collection if she has [latex]41[/latex] succulents.

Solution:

Succulents = six more than five times (tropical plants)

succulents = [latex]6+5(t)[/latex]

[latex]41=6+5t[/latex]

[latex]35=5t[/latex]

[latex]7=t[/latex]

Jessa owns 7 tropical plants.

	[latex]a--6=--8[/latex]
Add [latex]6[/latex] to each side to undo the subtraction.	[latex]a--6\color{red}{+6}=--8\color{red}{+6}[/latex]
Simplify.	[latex]a=--2[/latex]
Check the result by substituting [latex]-2[/latex] into the original equation:	[latex]a--6=--8[/latex]
Substitute [latex]-2[/latex] for [latex]a[/latex]	[latex]--2--6\stackrel{?}{=}--8[/latex]
	[latex]--8=--8\quad\checkmark[/latex]

	[latex]7x=--49[/latex]
Divide each side by [latex]7[/latex].	[latex]\Large{\frac{7x}{\color{red}{7}}=\frac{--49}{\color{red}{7}}}[/latex]
Simplify.	[latex]x=--7[/latex]

	[latex]7x=-49[/latex]
Substitute [latex]−7[/latex] for x.	[latex]7\left(-7\right)\stackrel{?}{=}-49[/latex]
	[latex]-49=-49\quad\checkmark[/latex]

	[latex]--3y=63[/latex]
Divide each side by [latex]−3[/latex].	[latex]\Large{\frac{--3y}{\color{red}{--3}}=\frac{63}{\color{red}{--3}}}[/latex]
Simplify	[latex]y=--21[/latex]

	[latex]-3y=63[/latex]
Substitute [latex]−21[/latex] for y.	[latex]-3\left(-21\right)\stackrel{?}{=}63[/latex]
	[latex]63=63\quad\checkmark[/latex]

Module 1: Integers