Lesson 2.1: Solve Equations Using the Subtraction and Addition Properties of Equality

New Concept

This lesson introduces the core principle of balancing equations. You'll use the Addition and Subtraction Properties of Equality to isolate a variable and find its value, a foundational skill for solving more complex algebraic problems.

What’s next

Soon, you'll walk through interactive examples and solve practice problems using these properties to find the value of the unknown variable.

Property

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

To determine whether a number is a solution to an equation.
Step 1. Substitute the number in for the variable in the equation.
Step 2. Simplify the expressions on both sides of the equation.
Step 3. Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.

Examples

• Is $y=4$ a solution to $5y - 3 = 17$? Substitute $y=4$: $5(4) - 3 = 20 - 3 = 17$. Since $17=17$, yes, it is a solution.
• Is $x=-3$ a solution to $2x + 8 = x+4$? Substitute $x=-3$: $2(-3) + 8 = 2$ and $-3+4=1$. Since $2 \neq 1$, it is not a solution.
• Is $a = \frac{1}{2}$ a solution to $8a - 1 = 3$? Substitute $a=\frac{1}{2}$: $8(\frac{1}{2}) - 1 = 4 - 1 = 3$. Since $3=3$, yes, it is a solution.

Property

For any numbers $a$, $b$, and $c$,

When you subtract the same quantity from both sides of an equation, you still have equality.

Examples

• To solve $x + 10 = 25$, subtract 10 from both sides: $x + 10 - 10 = 25 - 10$, which simplifies to $x = 15$.
• To solve $y + 4.2 = 8.7$, subtract 4.2 from both sides: $y + 4.2 - 4.2 = 8.7 - 4.2$, which simplifies to $y = 4.5$.
• To solve $z + 18 = 5$, subtract 18 from both sides: $z + 18 - 18 = 5 - 18$, which simplifies to $z = -13$.

Explanation

This property is like removing the same weight from both sides of a balanced scale. The scale stays balanced. We use this to 'undo' addition and isolate the variable to find its value.

Property

For any numbers $a$, $b$, and $c$,

When you add the same quantity to both sides of an equation, you still have equality.

Examples

• To solve $p - 8 = 14$, add 8 to both sides: $p - 8 + 8 = 14 + 8$, which simplifies to $p = 22$.
• To solve $q - \frac{1}{5} = \frac{3}{5}$, add $\frac{1}{5}$ to both sides: $q - \frac{1}{5} + \frac{1}{5} = \frac{3}{5} + \frac{1}{5}$, which simplifies to $q = \frac{4}{5}$.
• To solve $r - 5.5 = -10$, add 5.5 to both sides: $r - 5.5 + 5.5 = -10 + 5.5$, which simplifies to $r = -4.5$.

Explanation

This property helps you 'undo' subtraction in an equation. By adding the same number to both sides, you keep the equation balanced and get the variable all by itself.

Property

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 you try to isolate the variable. Simplify one side of the equation at a time.

Examples

• Solve $10x - 3 - 9x = 12$. First, combine like terms on the left: $(10x - 9x) - 3 = 12$, which gives $x - 3 = 12$. Then add 3 to both sides to get $x = 15$.
• Solve $4(z + 2) - 3z = 11$. First, distribute the 4: $4z + 8 - 3z = 11$. Combine like terms: $z + 8 = 11$. Then subtract 8 from both sides to get $z = 3$.
• Solve $2y + 5 + 3y = 4y + 9$. Combine terms on the left: $5y + 5 = 4y + 9$. Subtract $4y$ from both sides: $y + 5 = 9$. Then subtract 5 to get $y = 4$.

Explanation

It's like tidying up before solving a puzzle. Combine all like terms on the left side, then do the same on the right. This makes the equation much simpler and easier to solve correctly.

Property

To translate an English sentence to an algebraic equation:
Step 1. Locate the “equals” word(s). Translate to an equals 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

• Translate and solve: "Twenty more than a number $n$ is 35." This translates to $n + 20 = 35$. Subtracting 20 from both sides gives $n = 15$.
• Translate and solve: "The difference of $5y$ and $4y$ is -8." This translates to $5y - 4y = -8$. Simplifying the left side gives $y = -8$.
• Translate and solve: "A number $k$ decreased by 6 equals 11." This translates to $k - 6 = 11$. Adding 6 to both sides gives $k = 17$.

Explanation

This skill is like being a language translator, but for math! You look for clue words in a sentence to convert it into an equation. Words like 'is', 'gives', or 'results in' usually mean 'equals'.