Review of solving single-variable linear equations (including those with fractions and decimals)

TL;DR

You'll master solving equations like 3x + 5 = 14 and trickier ones with fractions like (2x - 1)/3 = 7. The goal is always to isolate x by undoing operations in reverse order. This foundation is crucial before tackling systems of equations.

1. The Mental Model

Think of an equation as a balance scale that must stay equal on both sides. Whatever you do to one side, you must do to the other to keep it balanced. Your mission is to get the variable alone on one side by "undoing" everything around it. That's the whole idea.

2. The Core Material

Basic Linear Equations

Start with the simplest form: ax + b = c. Your strategy is always the same - work backwards through the order of operations. If the equation says "multiply x by 3, then add 5," you'll subtract 5 first, then divide by 3.

Let's see this with 3x + 5 = 14:
- Subtract 5 from both sides: 3x = 9
- Divide both sides by 3: x = 3

Always check your answer by substituting back: 3(3) + 5 = 9 + 5 = 14 ✓

For equations where the variable appears on both sides like 5x - 3 = 2x + 9, collect all x terms on one side first:
- Subtract 2x from both sides: 3x - 3 = 9
- Add 3 to both sides: 3x = 12
- Divide by 3: x = 4

Equations with Fractions

Fractions make equations look scarier, but you have two solid approaches. You can either work with the fractions directly or clear them by multiplying through by the least common denominator (LCD).

For something like (2x - 1)/3 = 7, the direct approach works well:
- Multiply both sides by 3: 2x - 1 = 21
- Add 1 to both sides: 2x = 22
- Divide by 2: x = 11

But when you have multiple fractions like x/2 + x/3 = 10, clearing denominators is cleaner. The LCD of 2 and 3 is 6, so multiply everything by 6:
- 6(x/2) + 6(x/3) = 6(10)
- 3x + 2x = 60
- 5x = 60
- x = 12

Remember to distribute that LCD to every term, including constants.

Equations with Decimals

You can work with decimals directly, but it's often easier to clear them by multiplying by powers of 10. For 0.5x + 0.25 = 1.75, multiply everything by 100 to clear two decimal places:
- 50x + 25 = 175
- 50x = 150
- x = 3

The key is choosing the right power of 10. Look at the decimal with the most places and use that power.

graph TD
    A["Linear Equation"] --> B{"Fractions or Decimals?"}
    B -->|No| C["Use basic steps: undo operations"]
    B -->|Yes| D{"Clear them or work directly?"}
    D --> E["Clear: multiply by LCD/power of 10"]
    D --> F["Direct: careful arithmetic"]
    E --> C
    F --> C
    C --> G["Check answer by substitution"]

3. Worked Example

Let's solve this equation step by step: (3x + 2)/4 - x/6 = 5

First, I'll clear the fractions. The LCD of 4 and 6 is 12, so I'll multiply every term by 12:

12 · (3x + 2)/4 - 12 · x/6 = 12 · 5

This gives me:
3(3x + 2) - 2x = 60

Now I'll distribute the 3:
9x + 6 - 2x = 60

Combine like terms:
7x + 6 = 60

Subtract 6 from both sides:
7x = 54

Divide by 7:
x = 54/7

Let me check this answer by substituting back into the original equation:
(3 · 54/7 + 2)/4 - (54/7)/6 = (162/7 + 14/7)/4 - 9/7 = (176/7)/4 - 9/7 = 44/7 - 9/7 = 35/7 = 5 ✓

4. Key Takeaways

4.1 Most Important Concepts

• Balance principle: Whatever you do to one side of an equation, do to the other side to maintain equality.
• Reverse order of operations: Undo addition/subtraction first, then multiplication/division.
• Collecting like terms: Group all variable terms on one side and constants on the other.
• Fraction clearing: Multiply through by the LCD to eliminate fractions and simplify calculations.
• Decimal clearing: Multiply by appropriate powers of 10 to work with whole numbers instead.
• Always check: Substitute your answer back into the original equation to verify it's correct.
• Distribute carefully: When clearing fractions or decimals, multiply every single term, not just some.

4.2 Common Misconceptions

• "I only multiply one side by the LCD" - You must multiply both sides of the equation to maintain balance.
• "I can ignore the denominator when distributing" - You must multiply the entire numerator, including each term inside parentheses.
• "Decimals are harder than fractions" - Often decimals are actually easier to clear than complex fractions.
• "I should always clear fractions first" - Sometimes working directly with simple fractions is faster than clearing them.

4.3 Compare & Contrast

5. Now Try It

Solve these three equations using different approaches: (1) 2x/5 + 3 = 11, (2) x/4 - x/6 = 3, and (3) 0.3x + 1.2 = 2.7. For the first, work directly with the fraction. For the second, clear fractions using LCD. For the third, clear decimals. Success looks like getting x = 20, x = 36, and x = 5 respectively, with each answer checking out when substituted back.