3.5 Solve equations using integers; the division property of

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.

Determine whether each of the following is a solution of $2x-5=\mathrm{-13}\text{:}$

1. ⓐ $x=4$
2. ⓑ $x=\mathrm{-4}$
3. ⓒ $x=\mathrm{-9}.$

Solution

ⓐ Substitute 4 for x in the equation to determine if it is true.
Multiply.
Subtract.

Since $x=4$ does not result in a true equation, $4$ is not a solution to the equation.

ⓑ Substitute −4 for x in the equation to determine if it is true.
Multiply.
Subtract.

Since $x=\mathrm{-4}$ results in a true equation, $\mathrm{-4}$ is a solution to the equation.

ⓒ Substitute −9 for x in the equation to determine if it is true.
Substitute −9 for x.
Multiply.
Subtract.

Since $x=\mathrm{-9}$ does not result in a true equation, $\mathrm{-9}$ is not a solution to the equation.

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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.

Solve: $y+9=5.$

Solution

Subtract 9 from each side to undo the addition.
Simplify.

Check the result by substituting $\mathrm{-4}$ into the original equation.

	$y+9=5$
Substitute −4 for y	$\mathrm{-4}+9\stackrel{?}{=}5$
	$5=5✓$

Since $y=\mathrm{-4}$ makes $y+9=5$ a true statement, we found the solution to this equation.

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