Solve Equations Using Integers; The Division Property of Equality

Determine whether a number is a solution of an equation

A solution of an equation is a value of the variable that makes a true statement when substituted into the equation. Now that we’ve worked with integers, we can find integer solutions to equations. The steps are the same whether the solution turns out to be a whole number or an integer.

Example. Determine whether each of the following is a solution of $2x - 5 = -13$: (a) $x=4$, (b) $x=-4$, (c) $x=-9$.

(a) Substitute $4$ for $x$: $2(4)-5 = 8-5 = 3 \ne -13$. Since $x=4$ does not result in a true equation, $4$ is not a solution. (b) Substitute $-4$ for $x$: $2(-4)-5 = -8-5 = -13$. Since this is true, $-4$ is a solution. (c) Substitute $-9$ for $x$: $2(-9)-5 = -18-5 = -23 \ne -13$. Since this is not true, $-9$ is not a solution.

Solve equations with integers using the Addition and Subtraction Properties of Equality

We can use the same properties of equality we used with whole numbers to solve equations with integers: when you add or subtract the same quantity from both sides of an equation, you still have equality.

Example. Solve $y + 9 = 5$. Subtract $9$ from each side to undo the addition: $y+9-9 = 5-9$. Simplify: $y = -4$. Check by substituting $-4$ into the original equation: $-4+9 = 5$, and $5=5$ ✓.

Example. Solve $a - 6 = -8$. Add $6$ to each side to undo the subtraction: $a-6+6 = -8+6$. Simplify: $a = -2$. Check: $-2-6 = -8$ ✓.

Model the Division Property of Equality

All of the equations solved so far have had the form $x+a=b$ or $x-a=b$, where isolating the variable meant adding or subtracting a constant. Now let’s see how to solve equations that involve multiplication — where the variable is multiplied by a number.

Picture two identical envelopes, each containing the same unknown number of counters, sitting next to $6$ loose counters — the left side of the scale must equal the right side, but the counters in the envelopes are “hidden.” To find how many are in each envelope, separate the $6$ counters into $2$ equal groups: $6 \div 2 = 3$ counters in each envelope.

The equation modeling this is $2x = 6$. We divide both sides of the equation by $2$, the same way we split the counters into groups:

Does this check? $2 \cdot 3 = 6$ ✓ — three counters in each of two envelopes does equal six.

A second example: three identical envelopes balanced against $12$ counters models $3x=12$. Separating the $12$ counters into $3$ groups gives $4$ counters per envelope, since $12 \div 3 = 4$:

Check: $3 \cdot 4 = 12$ ✓.

Solve equations using the Division Property of Equality

These examples lead to the Division Property of Equality: when you divide both sides of an equation by any nonzero number, you still have equality.

Example. Solve $7x = -49$. To isolate $x$, undo the multiplication by dividing each side by $7$: $\tfrac{7x}{7} = \tfrac{-49}{7}$. Simplify: $x=-7$. Check: $7(-7) = -49$ ✓.

Example. Solve $-3y = 63$. Divide each side by $-3$: $\tfrac{-3y}{-3} = \tfrac{63}{-3}$. Simplify: $y=-21$. Check: $-3(-21) = 63$ ✓.

Translate word sentences to equations and solve

Now we’ll translate word sentences into equations with a variable, then solve them.

Example. Translate and solve: five more than $x$ is equal to $-3$. Translate: $x+5=-3$. Subtract $5$ from both sides: $x+5-5 = -3-5$. Simplify: $x=-8$. Check: $-8+5 = -3$ ✓.

Example. Translate and solve: the difference of $n$ and $6$ is $-10$. Translate: $n-6=-10$. Add $6$ to each side: $n-6+6 = -10+6$. Simplify: $n=-4$. Check: $-4-6=-10$ ✓.

Example. Translate and solve: the number $108$ is the product of $-9$ and $y$. Translate: $108 = -9y$. Divide by $-9$: $\tfrac{108}{-9} = \tfrac{-9y}{-9}$. Simplify: $-12=y$. Check: $108 = -9(-12)$ ✓.

Key terms

solution — a value of a variable that makes an equation’s statement true. Subtraction/Addition Property of Equality — adding or subtracting the same quantity from both sides of an equation preserves equality. Division Property of Equality — dividing both sides of an equation by the same nonzero number preserves equality.

Practice

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 word sentences to equations and solve

This section is adapted from Prealgebra 2e, Section 3.5: Solve Equations Using Integers; The Division Property of Equality by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the envelopes-and-counters models as accessible inline graphics; condensed prose; omitted the Be Prepared quiz, Manipulative Mathematics callout, and media links; converted the practice problems (“Try Its”) into interactive exercises with instant feedback; and adapted selected end-of-section exercises into the interactive Practice block, with each multipart exercise expanded into one question per part.