Solving Linear Equations

What Are Equations?

An equation is a mathematical statement that shows two expressions are equal, connected by an equals sign (=). Think of an equation as a perfectly balanced scale: whatever is on the left side must equal what’s on the right side.

For example, in the equation x + 5 = 12, we’re saying “some number plus 5 equals 12.” The unknown number is represented by a variable, usually a letter like x, y, or n.

Key Vocabulary

• Variable: A letter or symbol representing an unknown number (like x, y, or n)
• Coefficient: The number multiplied by a variable (in 3x, the coefficient is 3)
• Constant: A number on its own (like 5, -2, or 100)
• Solution: The value of the variable that makes the equation true
• Inverse Operations: Opposite operations that undo each other

The Golden Rule: Keep It Balanced

The most important principle in solving equations is balance. Whatever you do to one side of the equation, you must do to the other side. This keeps the equation equal, just like keeping a scale balanced.

Think of it this way: If you have a balanced scale and you add 3 kg to the left side, you must add 3 kg to the right side to keep it balanced. The same applies to equations.

Inverse Operations: The Key to Solving

Inverse operations are mathematical opposites that undo each other. We use them to isolate the variable (get it by itself on one side of the equation).

The main inverse operation pairs are:

• Addition ↔ Subtraction: Adding 5 is undone by subtracting 5
• Multiplication ↔ Division: Multiplying by 3 is undone by dividing by 3

One-Step Equations

One-step equations require only one inverse operation to solve.

Type 1: Addition Equations

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

Pattern: x + a = b, so x = b - a

Example 1: Basic Addition

Solve: x + 7 = 15

Solution:

x + 7 = 15
x + 7 - 7 = 15 - 7    (Subtract 7 from both sides)
x = 8

Check: 8 + 7 = 15 ✓

Type 2: Subtraction Equations

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

Pattern: x - a = b, so x = b + a

Example 2: Basic Subtraction

Solve: x - 9 = 14

Solution:

x - 9 = 14
x - 9 + 9 = 14 + 9    (Add 9 to both sides)
x = 23

Check: 23 - 9 = 14 ✓

Type 3: Multiplication Equations

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

Pattern: ax = b, so x = b ÷ a

Example 3: Basic Multiplication

Solve: 5x = 35

Solution:

5x = 35
5x ÷ 5 = 35 ÷ 5    (Divide both sides by 5)
x = 7

Check: 5 × 7 = 35 ✓

Type 4: Division Equations

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

Pattern: x ÷ a = b, so x = b × a

Example 4: Basic Division

Solve: x ÷ 4 = 9

Solution:

x ÷ 4 = 9
x ÷ 4 × 4 = 9 × 4    (Multiply both sides by 4)
x = 36

Check: 36 ÷ 4 = 9 ✓

Two-Step Equations

Two-step equations require two inverse operations to solve. The key is to work in reverse order of operations:

1. First, undo addition or subtraction
2. Then, undo multiplication or division

The Standard Form: ax + b = c

Most two-step equations follow this pattern, where you need to:

1. Subtract b from both sides (or add if it’s negative)
2. Divide both sides by a (or multiply if it’s a fraction)

Example 5: Standard Two-Step

Solve: 3x + 8 = 23

Solution:

3x + 8 = 23
3x + 8 - 8 = 23 - 8    (Step 1: Subtract 8 from both sides)
3x = 15
3x ÷ 3 = 15 ÷ 3        (Step 2: Divide both sides by 3)
x = 5

Check: 3(5) + 8 = 15 + 8 = 23 ✓

Example 6: Two-Step with Subtraction

Solve: 2x - 6 = 10

Solution:

2x - 6 = 10
2x - 6 + 6 = 10 + 6    (Step 1: Add 6 to both sides)
2x = 16
2x ÷ 2 = 16 ÷ 2        (Step 2: Divide both sides by 2)
x = 8

Check: 2(8) - 6 = 16 - 6 = 10 ✓

Example 7: Two-Step with Larger Numbers

Solve: 7x + 15 = 50

Solution:

7x + 15 = 50
7x + 15 - 15 = 50 - 15    (Subtract 15 from both sides)
7x = 35
7x ÷ 7 = 35 ÷ 7           (Divide both sides by 7)
x = 5

Check: 7(5) + 15 = 35 + 15 = 50 ✓

Example 8: Two-Step with Negative Coefficient

Solve: -4x + 12 = 0

Solution:

-4x + 12 = 0
-4x + 12 - 12 = 0 - 12    (Subtract 12 from both sides)
-4x = -12
-4x ÷ (-4) = -12 ÷ (-4)   (Divide both sides by -4)
x = 3

Check: -4(3) + 12 = -12 + 12 = 0 ✓

Equations with Variables on Both Sides

Sometimes variables appear on both sides of the equation. The strategy is to collect all variable terms on one side and all constant terms on the other.

Example 9: Variables on Both Sides

Solve: 5x + 3 = 2x + 15

Solution:

5x + 3 = 2x + 15
5x - 2x + 3 = 2x - 2x + 15    (Subtract 2x from both sides)
3x + 3 = 15
3x + 3 - 3 = 15 - 3           (Subtract 3 from both sides)
3x = 12
3x ÷ 3 = 12 ÷ 3               (Divide both sides by 3)
x = 4

Check: 5(4) + 3 = 20 + 3 = 23, and 2(4) + 15 = 8 + 15 = 23 ✓

Step-by-Step Strategy for Solving Equations

Follow these steps for any linear equation:

1. Simplify both sides if needed (combine like terms)
2. Move variable terms to one side using inverse operations
3. Move constant terms to the other side using inverse operations
4. Isolate the variable by dividing or multiplying
5. Check your solution by substituting back into the original equation

Common Errors and How to Avoid Them

Practice Problems

One-Step Equations

1. Solve: x + 15 = 28 Answer: x = 13

2. Solve: x - 9 = 17 Answer: x = 26

3. Solve: 6x = 42 Answer: x = 7

4. Solve: x ÷ 5 = 8 Answer: x = 40

5. Solve: x + 23 = 50 Answer: x = 27

6. Solve: 9x = 72 Answer: x = 8

Two-Step Equations

7. Solve: 2x + 5 = 19 Answer: x = 7

8. Solve: 3x - 7 = 14 Answer: x = 7

9. Solve: 5x + 12 = 37 Answer: x = 5

10. Solve: 4x - 9 = 27 Answer: x = 9

11. Solve: 6x + 8 = 44 Answer: x = 6

12. Solve: 7x - 15 = 20 Answer: x = 5

Challenge Problems

13. Solve: 8x + 25 = 73 Answer: x = 6

14. Solve: -3x + 18 = 6 Answer: x = 4

15. Solve: 5x + 3 = 3x + 19 Answer: x = 8

16. Solve: 12 - 2x = 4 Answer: x = 4

17. Solve: 10x - 35 = 65 Answer: x = 10

18. Solve: 4x + 7 = 2x + 23 Answer: x = 8

Real-World Applications

Application 1: Shopping Budget

Problem: You have $85 to spend on books. Each book costs$12, and you want to buy a bookmark for $13. How many books can you buy?

Solution: Let x = number of books Equation: 12x + 13 = 85

12x + 13 = 85
12x = 72
x = 6

You can buy 6 books.

Application 2: Temperature Conversion

Problem: The formula to convert Celsius to Fahrenheit is F = (9/5)C + 32. If the temperature is 77°F, what is it in Celsius?

Solution:

77 = (9/5)C + 32
77 - 32 = (9/5)C
45 = (9/5)C
45 × (5/9) = C
25 = C

The temperature is 25°C.

Application 3: Savings Plan

Problem: Emily wants to save $450 for a new tablet. She already has$120 saved and plans to save $15 per week. How many weeks until she reaches her goal?

Solution: Let x = number of weeks Equation: 120 + 15x = 450

120 + 15x = 450
15x = 330
x = 22

Emily needs to save for 22 weeks.

Application 4: Age Problems

Problem: Sarah is 4 years older than twice her brother’s age. If Sarah is 18, how old is her brother?

Solution: Let x = brother’s age Equation: 2x + 4 = 18

2x + 4 = 18
2x = 14
x = 7

Sarah’s brother is 7 years old.

Application 5: Perimeter Problems

Problem: A rectangle has a perimeter of 46 cm. Its length is 13 cm. What is its width?

Solution: Perimeter formula: P = 2l + 2w Let w = width

46 = 2(13) + 2w
46 = 26 + 2w
20 = 2w
w = 10

The width is 10 cm.

Why This Matters

Understanding how to solve equations is fundamental to algebra and mathematics beyond middle school. Equations allow us to:

• Model real situations with mathematical language
• Find unknown quantities in science, engineering, and everyday life
• Make predictions and solve problems systematically
• Think logically about relationships between quantities
• Build a foundation for advanced mathematics in high school and beyond

From calculating discounts while shopping to predicting future savings, from converting recipes to solving physics problems, equation-solving skills are tools you’ll use throughout your life. Every time you think “I need to figure out what number makes this work,” you’re thinking algebraically.

Connection to Other Topics

Solving equations connects to many other mathematical concepts:

• Order of Operations: Understanding PEMDAS helps you simplify equations correctly
• Properties of Equality: Addition, subtraction, multiplication, and division properties justify each step
• Integer Operations: Working with positive and negative numbers in equations
• Fractions and Decimals: Equations can involve any type of number
• Graphing: Solutions to equations can be visualized on coordinate planes
• Functions: Equations are the foundation for understanding functions in high school

Tips for Success

1. Always write your work neatly: Show each step clearly so you can check your work
2. Check your answer: Substitute your solution back into the original equation
3. Keep the equation balanced: Whatever you do to one side, do to the other
4. Work in reverse order: Undo addition/subtraction first, then multiplication/division
5. Practice regularly: The more equations you solve, the more confident you’ll become
6. Learn from mistakes: If you get an answer wrong, figure out where you went wrong
7. Use estimation: Before solving, estimate what the answer might be
8. Stay organized: Line up equals signs vertically to see the steps clearly

Mastering equation solving takes practice, but with patience and persistence, you’ll develop a powerful mathematical skill that opens doors to advanced mathematics and real-world problem solving.