Solving One-Step Equations

Welcome to equation solving! An equation is different from an expression - it has an equal sign and asks you to find the value that makes it true. In this topic, you'll learn to solve one-step equations using inverse operations.

What Is an Equation?

An equation is a mathematical sentence that says two expressions are equal.

Examples:

• x + 5 = 12 (equation)
• 3n = 21 (equation)
• 2x + 7 (expression - no equal sign!)

The Goal: Find the value of the variable that makes the equation true.

What Does "Solve" Mean?

To solve an equation means to find the value of the variable that makes both sides equal.

Example: x + 3 = 10

• Try x = 5: 5 + 3 = 8 (not equal to 10) ✗
• Try x = 7: 7 + 3 = 10 (equal!) ✓
• Solution: x = 7

The Balance Method

Think of an equation like a balanced scale. Whatever you do to one side, you must do to the other side to keep it balanced!

Key Rule: If you add, subtract, multiply, or divide on one side, you MUST do the same to the other side.

Inverse Operations

Inverse operations are operations that "undo" each other.

Pairs of Inverses:

• Addition ↔ Subtraction
• Multiplication ↔ Division

How to use them:

• If a number is added, subtract to undo it
• If a number is subtracted, add to undo it
• If a number is multiplied, divide to undo it
• If a number is divided, multiply to undo it

Addition Equations

When a number is added to the variable, subtract that number from both sides.

Example 1: Basic Addition

Solve: x + 7 = 15

Step 1: Identify what's being done to x x has 7 added to it

Step 2: Use inverse operation (subtract 7) x + 7 - 7 = 15 - 7

Step 3: Simplify both sides x = 8

Step 4: Check your answer 8 + 7 = 15 ✓

Answer: x = 8

Example 2: Addition on Right

Solve: 12 = n + 5

Step 1: Subtract 5 from both sides 12 - 5 = n + 5 - 5

Step 2: Simplify 7 = n

Answer: n = 7 (or we can write x = 7)

Subtraction Equations

When a number is subtracted from the variable, add that number to both sides.

Example 1: Basic Subtraction

Solve: y - 4 = 10

Step 1: Identify what's being done to y y has 4 subtracted from it

Step 2: Use inverse operation (add 4) y - 4 + 4 = 10 + 4

Step 3: Simplify y = 14

Step 4: Check 14 - 4 = 10 ✓

Answer: y = 14

Example 2: Larger Numbers

Solve: a - 23 = 45

Step 1: Add 23 to both sides a - 23 + 23 = 45 + 23

Step 2: Simplify a = 68

Step 3: Check 68 - 23 = 45 ✓

Answer: a = 68

Multiplication Equations

When a variable is multiplied by a number, divide both sides by that number.

Example 1: Basic Multiplication

Solve: 3x = 21

Step 1: Identify what's being done to x x is multiplied by 3

Step 2: Use inverse operation (divide by 3) 3x ÷ 3 = 21 ÷ 3

Step 3: Simplify x = 7

Step 4: Check 3(7) = 21 ✓

Answer: x = 7

Example 2: Larger Coefficient

Solve: 8n = 72

Step 1: Divide both sides by 8 8n ÷ 8 = 72 ÷ 8

Step 2: Simplify n = 9

Step 3: Check 8(9) = 72 ✓

Answer: n = 9

Example 3: Variable on Right

Solve: 45 = 5y

Step 1: Divide both sides by 5 45 ÷ 5 = 5y ÷ 5

Step 2: Simplify 9 = y

Answer: y = 9

Division Equations

When a variable is divided by a number, multiply both sides by that number.

Example 1: Basic Division

Solve: x/4 = 5

Step 1: Identify what's being done to x x is divided by 4

Step 2: Use inverse operation (multiply by 4) (x/4) × 4 = 5 × 4

Step 3: Simplify x = 20

Step 4: Check 20/4 = 5 ✓

Answer: x = 20

Example 2: Fraction Form

Solve: n/6 = 3

Step 1: Multiply both sides by 6 (n/6) × 6 = 3 × 6

Step 2: Simplify n = 18

Step 3: Check 18/6 = 3 ✓

Answer: n = 18

Example 3: Larger Divisor

Solve: y/12 = 7

Step 1: Multiply both sides by 12 (y/12) × 12 = 7 × 12

Step 2: Simplify y = 84

Answer: y = 84

Equations with Negative Numbers

Addition with Negatives

Solve: x + (-5) = 8 or written as: x - 5 = 8

Step 1: Add 5 to both sides x - 5 + 5 = 8 + 5

Step 2: Simplify x = 13

Answer: x = 13

Subtraction with Negatives

Solve: n - (-3) = 10 or: n + 3 = 10

Step 1: Subtract 3 from both sides n + 3 - 3 = 10 - 3

Step 2: Simplify n = 7

Answer: n = 7

Multiplication with Negatives

Solve: -4x = 28

Step 1: Divide both sides by -4 -4x ÷ (-4) = 28 ÷ (-4)

Step 2: Simplify (positive ÷ negative = negative) x = -7

Step 3: Check -4(-7) = 28 ✓

Answer: x = -7

Equations with Fractions and Decimals

Fraction Solutions

Solve: 2x = 5

Step 1: Divide both sides by 2 2x ÷ 2 = 5 ÷ 2

Step 2: Simplify x = 5/2 or 2.5

Answer: x = 5/2 (or 2.5 or 2 1/2)

Decimal Equations

Solve: x + 3.5 = 10

Step 1: Subtract 3.5 from both sides x + 3.5 - 3.5 = 10 - 3.5

Step 2: Simplify x = 6.5

Answer: x = 6.5

Real-World Applications

Shopping Problem

Problem: You have $50 and spend$x on a shirt. You have $27 left. How much did the shirt cost?

Equation: 50 - x = 27

Solution: 50 - x = 27 50 - x - 50 = 27 - 50 -x = -23 x = 23

Answer: The shirt cost $23

Distance Problem

Problem: A car travels at 60 mph. After how many hours will it have traveled 180 miles?

Equation: 60t = 180

Solution: 60t = 180 60t ÷ 60 = 180 ÷ 60 t = 3

Answer: 3 hours

Sharing Problem

Problem: You split a prize of x dollars equally among 4 friends. Each person gets $15. What was the total prize?

Equation: x/4 = 15

Solution: x/4 = 15 (x/4) × 4 = 15 × 4 x = 60

Answer: $60 total prize

Checking Your Solution

Always check by substituting your answer back into the original equation!

Example: Solve x + 8 = 15

Solution: x = 7

Check: 7 + 8 = 15 ✓ (True, so x = 7 is correct!)

If the check doesn't work, you made an error and need to try again.

Common Mistakes to Avoid

Mistake 1: Not doing the same thing to both sides Wrong: x + 5 = 12 → x = 12 Right: x + 5 = 12 → x + 5 - 5 = 12 - 5 → x = 7

Mistake 2: Using the wrong inverse operation Wrong: 3x = 15 → 3x + 3 = 15 + 3 Right: 3x = 15 → 3x ÷ 3 = 15 ÷ 3 → x = 5

Mistake 3: Sign errors with negatives Wrong: -2x = 10 → x = -20 Right: -2x = 10 → x = -5 (divide both by -2)

Mistake 4: Forgetting to check Always substitute your answer back to verify!

Mistake 5: Confusing expressions and equations Expression: 2x + 5 (can't solve, no equal sign) Equation: 2x + 5 = 15 (can solve for x)

Strategy for Solving

Step 1: Identify the operation What is being done to the variable? (added, subtracted, multiplied, divided)

Step 2: Choose the inverse operation Addition → Subtract Subtraction → Add Multiplication → Divide Division → Multiply

Step 3: Apply to both sides Keep the equation balanced!

Step 4: Simplify Perform the arithmetic

Step 5: Check your answer Substitute back into the original equation

Writing Equations from Words

Example 1: Addition

Words: "A number increased by 8 equals 20" Equation: n + 8 = 20 Solution: n = 12

Example 2: Multiplication

Words: "Five times a number is 35" Equation: 5x = 35 Solution: x = 7

Example 3: Division

Words: "A number divided by 3 gives 9" Equation: n/3 = 9 Solution: n = 27

Connection to Two-Step Equations

One-step equations are the foundation! Soon you'll solve equations that require two steps:

One-step: 2x = 10 (just divide) Two-step: 2x + 3 = 13 (subtract, then divide)

Master one-step equations first, and two-step will be easy!

Why Equations Matter

• Solving real problems: Find unknown values in everyday situations
• Science formulas: Rearrange formulas to solve for different variables
• Financial planning: Budget problems, savings calculations
• Construction and design: Finding measurements and dimensions
• Computer science: Programming uses equations constantly

Understanding equations is one of the most powerful tools in all of mathematics. Once you can solve equations, you can answer questions like "How much?" "How many?" and "How long?" in countless real-world situations!