Solving Linear Equations

What is a Linear Equation?

A linear equation is an equation where the variable appears to the first power (no exponents like x² or x³).

Examples of linear equations:

• x + 5 = 12
• 3x - 7 = 14
• 2(x + 3) = 10
• 4x + 1 = 2x + 9

Not linear equations:

• x² + 5 = 12 (has x²)
• 1/x = 3 (x in denominator)
• √x = 4 (has square root)

Standard Form

ax + b = c where a, b, and c are constants

Goal: Isolate x on one side of the equation

One-Step Equations

Addition/Subtraction:

Example 1: x + 7 = 15

Subtract 7 from both sides: x + 7 - 7 = 15 - 7 x = 8

Check: 8 + 7 = 15 ✓

Example 2: x - 4 = 11

Add 4 to both sides: x - 4 + 4 = 11 + 4 x = 15

Check: 15 - 4 = 11 ✓

Multiplication/Division:

Example 3: 3x = 21

Divide both sides by 3: 3x/3 = 21/3 x = 7

Check: 3(7) = 21 ✓

Example 4: x/5 = 8

Multiply both sides by 5: (x/5) · 5 = 8 · 5 x = 40

Check: 40/5 = 8 ✓

Two-Step Equations

Form: ax + b = c

Steps:

1. Add or subtract to move constant term
2. Multiply or divide to isolate variable

Example 1: 2x + 5 = 17

Step 1: Subtract 5 2x + 5 - 5 = 17 - 5 2x = 12

Step 2: Divide by 2 2x/2 = 12/2 x = 6

Check: 2(6) + 5 = 12 + 5 = 17 ✓

Example 2: 3x - 8 = 13

Step 1: Add 8 3x - 8 + 8 = 13 + 8 3x = 21

Step 2: Divide by 3 x = 7

Check: 3(7) - 8 = 21 - 8 = 13 ✓

Example 3: x/4 + 3 = 10

Step 1: Subtract 3 x/4 = 7

Step 2: Multiply by 4 x = 28

Check: 28/4 + 3 = 7 + 3 = 10 ✓

Example 4: -5x + 2 = -18

Step 1: Subtract 2 -5x = -20

Step 2: Divide by -5 x = 4

Check: -5(4) + 2 = -20 + 2 = -18 ✓

Multi-Step Equations

With parentheses - use distributive property:

Example 1: 3(x + 4) = 21

Distribute 3: 3x + 12 = 21

Subtract 12: 3x = 9

Divide by 3: x = 3

Check: 3(3 + 4) = 3(7) = 21 ✓

Example 2: 2(x - 5) + 3 = 13

Distribute 2: 2x - 10 + 3 = 13

Combine like terms: 2x - 7 = 13

Add 7: 2x = 20

Divide by 2: x = 10

Check: 2(10 - 5) + 3 = 2(5) + 3 = 13 ✓

Example 3: -4(2x + 1) = 20

Distribute -4: -8x - 4 = 20

Add 4: -8x = 24

Divide by -8: x = -3

Check: -4(2(-3) + 1) = -4(-6 + 1) = -4(-5) = 20 ✓

Variables on Both Sides

Strategy: Collect all variable terms on one side, constants on the other

Example 1: 5x + 3 = 2x + 12

Subtract 2x from both sides: 3x + 3 = 12

Subtract 3: 3x = 9

Divide by 3: x = 3

Check: 5(3) + 3 = 15 + 3 = 18, and 2(3) + 12 = 6 + 12 = 18 ✓

Example 2: 7x - 4 = 3x + 8

Subtract 3x: 4x - 4 = 8

Add 4: 4x = 12

Divide by 4: x = 3

Check: 7(3) - 4 = 17, and 3(3) + 8 = 17 ✓

Example 3: 2x + 9 = 6x - 3

Subtract 2x: 9 = 4x - 3

Add 3: 12 = 4x

Divide by 4: x = 3

Example 4: 8 - 3x = 2x + 13

Add 3x: 8 = 5x + 13

Subtract 13: -5 = 5x

Divide by 5: x = -1

Check: 8 - 3(-1) = 8 + 3 = 11, and 2(-1) + 13 = -2 + 13 = 11 ✓

Equations with Fractions

Method 1: Clear fractions by multiplying by LCD

Example 1: (x/3) + 2 = 5

Method 1 - Clear fraction: Multiply everything by 3: 3 · (x/3) + 3 · 2 = 3 · 5 x + 6 = 15 x = 9

Method 2 - Work with fraction: Subtract 2: x/3 = 3 Multiply by 3: x = 9

Example 2: (2x/5) - 1 = 3

Multiply by 5: 5 · (2x/5) - 5 · 1 = 5 · 3 2x - 5 = 15 2x = 20 x = 10

Example 3: (x/2) + (x/3) = 5

LCD = 6, multiply everything by 6: 6 · (x/2) + 6 · (x/3) = 6 · 5 3x + 2x = 30 5x = 30 x = 6

Check: 6/2 + 6/3 = 3 + 2 = 5 ✓

Example 4: (x + 2)/4 = 3

Multiply by 4: x + 2 = 12 x = 10

Equations with Decimals

Method: Clear decimals by multiplying by power of 10

Example 1: 0.5x + 1.2 = 3.7

Multiply by 10 (move decimal 1 place): 5x + 12 = 37 5x = 25 x = 5

Check: 0.5(5) + 1.2 = 2.5 + 1.2 = 3.7 ✓

Example 2: 0.25x - 0.5 = 1.75

Multiply by 100 (move decimal 2 places): 25x - 50 = 175 25x = 225 x = 9

Example 3: 1.5x + 2.4 = 0.3x + 8.4

Multiply by 10: 15x + 24 = 3x + 84 12x = 60 x = 5

Combining Like Terms First

Example 1: 3x + 2x - 4 = 11

Combine 3x + 2x: 5x - 4 = 11 5x = 15 x = 3

Example 2: 7x - 2x + 5 = 3x + 17

Combine 7x - 2x: 5x + 5 = 3x + 17

Subtract 3x: 2x + 5 = 17 2x = 12 x = 6

Example 3: 4(x + 2) + 3(x - 1) = 26

Distribute: 4x + 8 + 3x - 3 = 26

Combine like terms: 7x + 5 = 26 7x = 21 x = 3

Special Cases

Identity (infinite solutions):

Example: 2x + 4 = 2(x + 2)

Distribute: 2x + 4 = 2x + 4

Subtract 2x: 4 = 4 (always true!)

Solution: All real numbers (infinite solutions)

Contradiction (no solution):

Example: 3x + 5 = 3x + 8

Subtract 3x: 5 = 8 (never true!)

Solution: No solution (empty set)

Checking Your Answer

Why check? Catch arithmetic mistakes!

How to check:

1. Substitute answer back into original equation
2. Simplify both sides
3. Verify left side equals right side

Example: If x = 4 in 2x + 3 = 11

Check: 2(4) + 3 = 8 + 3 = 11 ✓

Real-World Applications

Example 1: Age Problem

Maria is 3 years older than twice John's age. If Maria is 23, how old is John?

Let x = John's age 2x + 3 = 23 2x = 20 x = 10

John is 10 years old.

Example 2: Shopping

You buy 3 shirts at the same price and pay 8 dollars for shipping. Total cost is 53 dollars. What is the price per shirt?

Let x = price per shirt 3x + 8 = 53 3x = 45 x = 15

Each shirt costs 15 dollars.

Example 3: Temperature Conversion

Convert 77°F to Celsius using C = (5/9)(F - 32)

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

Example 4: Perimeter

A rectangle has perimeter 40 feet. Length is 2 feet more than width. Find the width.

Let w = width, then length = w + 2 Perimeter: 2w + 2(w + 2) = 40 2w + 2w + 4 = 40 4w + 4 = 40 4w = 36 w = 9

Width is 9 feet, length is 11 feet.

Common Mistakes to Avoid

1. Not doing same operation to both sides WRONG: 2x + 3 = 11 → 2x = 11 - 3 (forgot to subtract from left!) CORRECT: 2x + 3 = 11 → 2x = 8

2. Sign errors with negatives Dividing both sides by -5: -5x = 20 → x = -4 (not +4!)

3. Not distributing to all terms 3(x + 4) = 3x + 12 (not 3x + 4!)

4. Forgetting to combine like terms first 5x + 2x = 14 → 7x = 14 → x = 2

5. Arithmetic mistakes Always check your answer!

Step-by-Step Strategy

General process:

1. Simplify each side (distribute, combine like terms)
2. Move variable terms to one side (add/subtract)
3. Move constant terms to other side (add/subtract)
4. Isolate variable (multiply/divide)
5. Check your answer

Remember: Whatever you do to one side, do to the other!

Quick Reference

One-step: Just add, subtract, multiply, or divide

Two-step: Move constant, then coefficient

With parentheses: Distribute first

Variables both sides: Collect all x's on one side

With fractions: Multiply by LCD to clear

With decimals: Multiply by 10, 100, or 1000

Always check: Substitute back into original equation

Practice Tips

• Start with simpler equations, build up to complex
• Show all steps even if you can do them mentally
• Write neatly to avoid sign errors
• Check every answer
• Use opposite operations (addition ↔ subtraction, multiplication ↔ division)
• Keep equations balanced (both sides equal)
• Practice with word problems to see real applications
• Don't rush - accuracy over speed
• If stuck, go back to basics
• Master this foundation - it appears everywhere in algebra!

Linear equations are the building blocks of algebra. Master them now and everything else becomes easier!