3.5 Solve equations using integers; the division property of  (Page 2/5)

Solve: $a-6=\mathrm{-8}$

Solution

Add 6 to each side to undo the subtraction.
Simplify.

Check the result by substituting $\mathrm{-2}$ into the original equation: $a-6=\mathrm{-8}.$

Substitute $\mathrm{-2}$ for $a$	$\mathrm{-2}-6\stackrel{?}{=}\mathrm{-8}$
	$\mathrm{-8}=\mathrm{-8}✓$

The solution to $a-6=\mathrm{-8}$ is $\mathrm{-2}.$

Since $a=\mathrm{-2}$ makes $a-6=\mathrm{-8}$ a true statement, we found the solution to this equation.

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Model the division property of equality

All of the equations we have solved so far have been of the form $x+a=b$ or $x-a=b.$ 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 [link] .

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 $2$ groups of the same size. So $6$ counters divided into $2$ groups means there must be $3$ counters in each group (since $6÷2=3).$

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

We can divide both sides of the equation by $2$ as we did with the envelopes and counters.

We found that each envelope contains $\text{3 counters.}$ Does this check? We know $2·3=6,$ so it works. Three counters in each of two envelopes does equal six.

[link] shows another example.

Now we have $3$ identical envelopes and $\text{12 counters.}$ How many counters are in each envelope? We have to separate the $\text{12 counters}$ into $\text{3 groups.}$ Since $12÷3=4,$ there must be $\text{4 counters}$ in each envelope. See [link] .

The equation that models the situation is $3x=12.$ We can divide both sides of the equation by $3.$

Does this check? It does because $3·4=12.$

Write an equation modeled by the envelopes and counters, and then solve it.

Solution

There are $\text{4 envelopes,}$ or $4$ unknown values, on the left that match the $\text{8 counters}$ on the right. Let’s call the unknown quantity in the envelopes $x.$

Write the equation.
Divide both sides by 4.
Simplify.

There are $\text{2 counters}$ in each envelope.

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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.

$\text{Solve:}7x=\mathrm{-49}.$

Solution

To isolate $x,$ we need to undo multiplication.

Divide each side by 7.
Simplify.

Check the solution.

	$7x=\mathrm{-49}$
Substitute −7 for x.	$7\left(\mathrm{-7}\right)\stackrel{?}{=}\mathrm{-49}$
	$\mathrm{-49}=\mathrm{-49}✓$

Therefore, $\mathrm{-7}$ is the solution to the equation.

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