Lesson 3: Solving Equations Using the Subtraction and Addition Properties of Equality

New Concept

This lesson introduces the foundation of solving algebraic equations: isolating the variable. You'll learn how the Addition and Subtraction Properties of Equality allow you to 'undo' operations and find the value that makes an equation true.

What’s next

Next, you'll put these properties into action with interactive examples, practice problems, and challenge questions to master solving equations.

Property

A solution to an equation is a value of a variable that makes a true statement when substituted into the equation. The process of finding the solution to an equation is called solving the equation.

Examples

• Is $x=6$ a solution to $x+9=15$? We substitute 6 for $x$ to get $6+9=15$. Since $15=15$ is a true statement, 6 is a solution.

• Is $d=2$ a solution to $18d-9=27$? We substitute 2 for $d$ to get $18(2)-9 = 36-9=27$. Since $27=27$ is true, 2 is a solution.

Property

For any numbers $a$, $b$, and $c$, if $a=b$, then $a-c=b-c$. When we subtract the same quantity from both sides of an equation, we still have equality. The goal is to isolate the variable.

Examples

• To solve $x+8=17$, we use the Subtraction Property of Equality to isolate $x$. Subtracting 8 from both sides gives $x+8-8=17-8$, which simplifies to $x=9$.

• To solve $a+2=18$, we get $a$ by itself by subtracting 2 from both sides. This gives $a+2-2=18-2$, which simplifies to $a=16$.

Property

For any numbers $a$, $b$, and $c$, if $a=b$, then $a+c=b+c$. We can add the same number to both sides of the equation without changing the equality.

Examples

• To solve $x-5=8$, we use the Addition Property of Equality to isolate $x$. Adding 5 to both sides gives $x-5+5=8+5$, which simplifies to $x=13$.

• To solve $y-3=19$, we get $y$ by itself by adding 3 to both sides. This gives $y-3+3=19+3$, which simplifies to $y=22$.

Property

An equation has an equal sign between two algebraic expressions. To translate a sentence into an equation, look for clue words that mean 'equals' (is, gives, was, etc.). Then, translate the phrases on each side of the 'equals' word into expressions.

Examples

• The sentence 'The sum of 6 and 12 is equal to 18' translates to the equation $6+12=18$.

• The sentence 'The product of 4 and 7 is equal to 28' translates to the equation $4 \cdot 7=28$.

Property

First, translate a word sentence into an algebraic equation. Then, solve the equation by using the Subtraction and Addition Properties of Equality to isolate the variable.

Examples

• 'Five more than $p$ is equal to 21' translates to $p+5=21$. Subtracting 5 from both sides gives $p=16$.

• 'The difference of $y$ and 9 is 20' translates to $y-9=20$. Adding 9 to both sides gives $y=29$.