Lesson 1: Solve Equations Using the Subtraction and Addition Properties of Equality

New Concept

To solve an equation, you must keep it balanced. This lesson introduces the Subtraction and Addition Properties of Equality—your tools for adding or subtracting from both sides to isolate the variable and find its value.

What’s next

Next, you'll see these properties in action with worked examples, then apply them yourself in a series of interactive practice problems.

Property

A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

Examples

• To check if $x=9$ is a solution for $x-2=7$, substitute 9 for $x$. This gives $9-2=7$, which simplifies to $7=7$. Since this is a true statement, $x=9$ is a solution.

• Is $y=10$ a solution for $y-4=6$? Substitute 10 for $y$ to get $10-4=6$. This simplifies to $6=6$, which is true, so $y=10$ is a solution.

Property

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 $a + c = b + c$.
When you add or subtract the same quantity from both sides of an equation, you still have equality.

Examples

• To solve $p - 7 = 12$, we use the Addition Property. Add 7 to both sides: $p - 7 + 7 = 12 + 7$, which simplifies to $p = 19$.

Property

Step 1. Locate the 'equals' word(s). Translate to an equal sign.
Step 2. Translate the words to the left of the 'equals' word(s) into an algebraic expression.
Step 3. Translate the words to the right of the 'equals' word(s) into an algebraic expression.

Examples

• 'Five more than $x$ is equal to 26' translates to the equation $x + 5 = 26$. The words 'is equal to' become the $=$ sign.

• 'The difference of $5p$ and $4p$ is 23' translates to $5p - 4p = 23$. The word 'is' becomes the $=$ sign.

Property

Step 1. Read the problem. Make sure you understand all the words and ideas.
Step 2. Identify what you are looking for.
Step 3. Name what you are looking for. Choose a variable to represent that quantity.
Step 4. Translate into an equation.
Step 5. Solve the equation using good algebra techniques.
Step 6. Check the answer in the problem and make sure it makes sense.
Step 7. Answer the question with a complete sentence.

Examples

• Problem: A number increased by 9 is 21. Find the number. Identify 'the number' as $n$. Translate: $n + 9 = 21$. Solve: $n = 12$. The number is 12.

• Problem: A shirt costs 25 dollars after a 10 dollar discount. What was the original price? Identify 'original price' as $p$. Translate: $p - 10 = 25$. Solve: $p = 35$. The original price was 35 dollars.