Lesson 19: Solving One-Step Equations by Adding or Subtracting

New Concept

An equation is a statement that uses an equal sign to show that two quantities are equal. To solve an equation, we use properties of equality to keep it balanced, like a scale.

Addition Property of Equality
If $a = b$, then $a + c = b + c$.

Subtraction Property of Equality
If $a = b$, then $a - c = b - c$.

What’s next

This lesson builds your foundation with one-step equations. You'll use inverse operations in worked examples, solve word problems, and check your answers for accuracy.

Property

You can add the same number to both sides of an equation and the statement will still be true. If $a = b$, then $a + c = b + c$.

Examples

• To solve $x - 5 = 12$, add 5 to both sides: $x - 5 + 5 = 12 + 5$, so $x = 17.$
• To solve $-10 = n - 4$, add 4 to both sides: $-10 + 4 = n - 4 + 4$, so $-6 = n.$

Explanation

Imagine a balanced scale. If you add a 5-pound weight to the left side, you must also add a 5-pound weight to the right to keep it from tipping over! This is how we isolate the variable in equations involving subtraction. We add the same number back to both sides to 'undo' the subtraction and find our answer.

Property

You can subtract the same number from both sides of an equation and the statement will still be true. If $a = b$, then $a - c = b - c$.

Examples

• To solve $k + 8 = 15$, subtract 8 from both sides: $k + 8 - 8 = 15 - 8$, so $k = 7.$
• To solve $-20 = p + 10$, subtract 10 from both sides: $-20 - 10 = p + 10 - 10$, so $-30 = p.$

Explanation

Let's go back to our trusty balanced scale. If you take away a 3-ounce cookie from the left side, you must take a 3-ounce cookie from the right side to maintain balance. This is the key to solving equations with addition. We subtract the same number from both sides to cancel out the addition and isolate our lonely variable.

Property

Inverse operations are operations that undo each other. For example: Addition $\longleftrightarrow$ Subtraction.

Examples

• $In x + 9 = 14$, use the inverse of addition (subtraction): $x = 14 - 9$, so $x = 5.$
• $In y - 7 = 3$, use the inverse of subtraction (addition): $y = 3 + 7$, so $y = 10.$

Explanation

Think of inverse operations as an 'undo' button for math. If an equation has addition, you use subtraction to cancel it out and get the variable by itself. If it has subtraction, you use addition! It is like putting on your shoes (addition) and taking them off (subtraction)—one action reverses the other to get you back where you started.