Solve Equations Using the Subtraction and Addition Properties of Equality

We began our work solving equations in previous chapters. It has been a while since we have seen an equation, so we will review some of the key concepts before we go any further.

We said that solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle.

In earlier sections, we listed the steps to determine if a value is a solution. We restate them here.

Example. Determine whether $y = \tfrac{3}{4}$ is a solution for $4y + 3 = 8y$.

Substitute $\tfrac{3}{4}$ for $y$:

Multiply:

Add:

Since $y = \tfrac{3}{4}$ results in a true equation, $\tfrac{3}{4}$ is a solution to the equation $4y + 3 = 8y$.

Solve equations using the Subtraction and Addition Properties of Equality

We introduced the Subtraction and Addition Properties of Equality earlier, using envelopes and counters to model an equation such as $x + 3 = 8$. The goal is to isolate the variable on one side of the equation — so we “took away” $3$ from both sides and found the solution $x = 5$.

Some people picture a balance scale when they solve equations: one mass on each side is balanced; two masses on each side is balanced; but one mass on one side and two masses on the other is unbalanced.

The quantities on both sides of the equal sign in an equation are equal, or balanced. Just as with the balance scale, whatever you do to one side of the equation you must also do to the other to keep it balanced.

When you add or subtract the same quantity from both sides of an equation, you still have equality. Let’s review how to use the Subtraction and Addition Properties of Equality to solve equations. We need to isolate the variable on one side of the equation, and we check our solutions by substituting the value into the equation to make sure we have a true statement.

Example. Solve: $x - 11 = -3$.

To isolate $x$, we undo the subtraction of $11$ by using the Addition Property of Equality — we “undo” the subtraction of $11$ by adding $11$ to each side:

Simplify:

Check: substitute $x = 8$ into $x - 11 = -3$:

Since $x = 8$ makes $x - 11 = -3$ a true statement, $8$ is a solution to the equation.

In that example, $11$ was subtracted from $x$, so we added $11$ to “undo” the subtraction. In the next example, we need to “undo” addition by using the Subtraction Property of Equality.

Example. Solve: $m + 4 = -5$.

Subtract $4$ from each side to “undo” the addition:

Simplify:

Check: substitute $m = -9$ into $m + 4 = -5$:

The solution to $m + 4 = -5$ is $m = -9$.

Now let’s review solving equations with fractions.

Example. Solve: $n - \tfrac{3}{8} = \tfrac{1}{2}$.

Use the Addition Property of Equality, then find the LCD to add the fractions on the right:

Simplify:

Check: substitute $n = \tfrac{7}{8}$ into the original equation:

The solution checks.

In previous sections we also solved equations that contained decimals. We’ll review one of those next.

Example. Solve: $a - 3.7 = 4.3$.

Use the Addition Property of Equality:

Add:

Check: substitute $a = 8$ into $a - 3.7 = 4.3$:

The solution checks.

Solve equations that need to be simplified

In the examples up to this point, we have been able to isolate the variable with just one operation. Many of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation before using the Subtraction or Addition Properties of Equality. You should always simplify as much as possible before trying to isolate the variable.

Example. Solve: $3x - 7 - 2x - 4 = 1$.

The left side of the equation has an expression that we should simplify before trying to isolate the variable. Rearrange the terms using the Commutative Property of Addition, then combine like terms:

Add $11$ to both sides to isolate $x$, then simplify:

Check by substituting $x = 12$ into the original equation:

The solution checks.

Some equations have parentheses that must be distributed before we can combine like terms.

Example. Solve: $3(n - 4) - 2n = -3$.

The left side has an expression we should simplify. Distribute on the left, use the Commutative Property to rearrange terms, then combine like terms:

Isolate $n$ using the Addition Property of Equality, then simplify:

Check by substituting $n = 9$ into the original equation:

The solution checks.

Sometimes both sides of the equation need to be simplified before we isolate the variable.

Example. Solve: $2(3k - 1) - 5k = -2 - 7$.

Both sides have expressions we should simplify first. Distribute on the left and combine the constants on the right, then use the Commutative Property to rearrange terms and combine like terms:

Undo the subtraction by using the Addition Property of Equality, then simplify:

Check by substituting $k = -7$ into the original equation:

The solution checks.

Translate an equation and solve

In previous chapters, we translated word sentences into equations. The first step is to look for the word (or words) that translate(s) to the equal sign — words like is, is equal to, is the same as, the result is, gives, was, and will be all translate to $=$.

Example. Translate and solve: five more than $x$ is equal to $26$.

Translate “five more than $x$” as $x + 5$ and “is equal to” as $=$:

Subtract $5$ from both sides, then simplify:

Check: is $26$ five more than $21$?

The solution checks.

Example. Translate and solve: the difference of $5p$ and $4p$ is $23$.

Translate “the difference of $5p$ and $4p$” as $5p - 4p$ and “is” as $=$:

Simplify the left side:

Check by substituting $p = 23$ into the original equation:

The solution checks.

Translate and solve applications

In most of the application problems we solved earlier, we were able to find the quantity we were looking for by simplifying an algebraic expression. Now we will use equations to solve application problems. We’ll start by restating the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve. When assigning a variable, choose a letter that reminds you of what you are looking for.

Example. The Robles family has two dogs, Buster and Chandler. Together, they weigh $71$ pounds. Chandler weighs $28$ pounds. How much does Buster weigh?

Identify what you are asked to find, and choose a variable to represent it: how much does Buster weigh? Let $b =$ Buster’s weight.

Write a sentence that gives the information to find it, then restate it including the given information: Buster’s weight plus Chandler’s weight equals $71$ pounds — Buster’s weight plus $28$ equals $71$.

Translate the sentence into an equation, using the variable $b$:

Solve the equation:

Check the answer in the problem: is $43$ pounds a reasonable weight for a dog? Yes. Does Buster’s weight plus Chandler’s weight equal $71$ pounds?

Write a complete sentence that answers the question: Buster weighs $43$ pounds.

Example. Shayla paid $\text{\textdollar}24{,}575$ for her new car. This was $\text{\textdollar}875$ less than the sticker price. What was the sticker price of the car?

Identify what you are asked to find: “What was the sticker price of the car?” Assign a variable: let $s =$ the sticker price of the car.

Write a sentence that gives the information to find it: $\text{\textdollar}24{,}575$ is $\text{\textdollar}875$ less than the sticker price — $24{,}575$ is $875$ less than $s$.

Translate into an equation:

Solve:

Check: is $875$ less than $25{,}450$ equal to $24{,}575$?

The sticker price was $\text{\textdollar}25{,}450$.

Key terms

solution of an equation — a value of a variable that makes a true statement when substituted into the equation. Subtraction Property of Equality — for all real numbers $a$, $b$, and $c$, if $a = b$, then $a - c = b - c$. Addition Property of Equality — for all real numbers $a$, $b$, and $c$, if $a = b$, then $a + c = b + c$.

This section is adapted from Prealgebra 2e, Section 8.1: Solve Equations Using the Subtraction and Addition Properties 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 balance-scale figure as an accessible inline graphic; omitted the Be Prepared quiz, Links to Literacy and media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback, rephrasing the “is v a solution” check as a yes/no question.