🔓 Solving Linear Equations: Unlock the Unknown!

Imagine you have a treasure chest locked with a secret code. The code is hidden inside a puzzle. Your job? Solve the puzzle to find the code and open the chest!

That’s exactly what solving equations is all about—finding the hidden number!

🎯 What is an Equation?

Think of an equation like a balanced seesaw on a playground.

   🧒        =        🧒
  ────────────────────────
         ▲

Both sides must be perfectly equal. If one side gets heavier, the seesaw tips!

An equation has:

• An equals sign (=) in the middle
• Something on the left side
• Something on the right side

Simple Example

x + 3 = 7

This says: “Some mystery number plus 3 equals 7.”

The mystery number has a name: we call it x (but you can call it anything—even 🌟).

Your mission: Find what x is hiding!

✅ Solutions and Verification

A solution is the number that makes the equation TRUE.

How to Check Your Answer (Verification)

1. Take your answer
2. Put it back into the equation
3. See if both sides match

Example: Is x = 4 the solution to x + 3 = 7?

Put 4 where x is:
4 + 3 = 7
  7   = 7  ✓ YES! Both sides match!

Try a wrong answer: Is x = 5 the solution?

Put 5 where x is:
5 + 3 = 7
  8   = 7  ✗ NO! 8 is not 7!

💡 Remember: Always check your answer by putting it back in!

🥇 Solving One-Step Equations

One-step means you only need ONE move to find x.

The Magic Rule: Do the OPPOSITE!

Whatever is happening to x, do the opposite to get x alone.

graph TD
    A[See what's with x] --> B{What operation?}
    B --> |Adding| C[Subtract it]
    B --> |Subtracting| D[Add it]
    B --> |Multiplying| E[Divide by it]
    B --> |Dividing| F[Multiply by it]

Example 1: Addition → Subtract

x + 5 = 12

5 is adding to x. Do the opposite: subtract 5 from both sides!

x + 5 - 5 = 12 - 5
    x     =  7

✅ Answer: x = 7

Example 2: Subtraction → Add

x - 4 = 10

4 is subtracting from x. Do the opposite: add 4 to both sides!

x - 4 + 4 = 10 + 4
    x     =  14

✅ Answer: x = 14

Example 3: Multiplication → Divide

3x = 15

(3x means 3 times x)

x is being multiplied by 3. Do the opposite: divide both sides by 3!

3x ÷ 3 = 15 ÷ 3
   x   =  5

✅ Answer: x = 5

Example 4: Division → Multiply

x ÷ 2 = 6

x is being divided by 2. Do the opposite: multiply both sides by 2!

(x ÷ 2) × 2 = 6 × 2
     x      = 12

✅ Answer: x = 12

🥈 Solving Two-Step Equations

Now we need TWO moves to free x from its cage!

The Order Matters: Undo in Reverse!

Think about getting dressed:

• First socks, then shoes

To undress:

• First shoes, then socks (reverse order!)

For equations: Undo addition/subtraction FIRST, then multiplication/division.

Example: Two Steps

2x + 3 = 11

Step 1: Undo the +3 (subtract 3 from both sides)

2x + 3 - 3 = 11 - 3
   2x      =  8

Step 2: Undo the ×2 (divide both sides by 2)

2x ÷ 2 = 8 ÷ 2
   x   = 4

✅ Answer: x = 4

Verify: 2(4) + 3 = 8 + 3 = 11 ✓

Another Example

x/4 - 2 = 5

Step 1: Undo the -2 (add 2 to both sides)

x/4 - 2 + 2 = 5 + 2
    x/4     = 7

Step 2: Undo the ÷4 (multiply both sides by 4)

(x/4) × 4 = 7 × 4
    x     = 28

✅ Answer: x = 28

🏆 Multi-Step Linear Equations

Sometimes x appears on BOTH sides of the equation. Don’t panic! We just need to gather all the x’s together first.

Strategy: Get All x’s on One Side

3x + 4 = x + 12

Step 1: Move x to one side (subtract x from both sides)

3x - x + 4 = x - x + 12
   2x + 4  =    12

Step 2: Undo the +4

2x + 4 - 4 = 12 - 4
   2x      =  8

Step 3: Undo the ×2

2x ÷ 2 = 8 ÷ 2
   x   = 4

✅ Answer: x = 4

Example with Parentheses

2(x + 3) = 14

Step 1: Distribute (multiply 2 by everything inside)

2×x + 2×3 = 14
  2x + 6  = 14

Step 2: Undo the +6

2x + 6 - 6 = 14 - 6
   2x      =  8

Step 3: Undo the ×2

2x ÷ 2 = 8 ÷ 2
   x   = 4

✅ Answer: x = 4

🍕 Equations with Fractions

Fractions might look scary, but there’s a super trick: Multiply by the denominator to make fractions disappear!

Example 1: Simple Fraction

x/3 = 5

Multiply both sides by 3:

(x/3) × 3 = 5 × 3
    x     = 15

✅ Answer: x = 15

Example 2: Fraction Coefficient

(2/3)x = 8

To undo multiplying by 2/3, multiply by its reciprocal (flip it: 3/2):

(3/2) × (2/3)x = 8 × (3/2)
       x       = 24/2
       x       = 12

✅ Answer: x = 12

Example 3: Fractions on Both Sides

x/2 + 1/4 = 3/4

Step 1: Find a common denominator (4) and multiply everything by 4:

4 × (x/2) + 4 × (1/4) = 4 × (3/4)
   2x     +     1     =     3

Step 2: Solve the regular equation:

2x + 1 - 1 = 3 - 1
   2x      = 2
    x      = 1

✅ Answer: x = 1

🔤 Literal Equations

A literal equation has more than one letter (like formulas you use in science or geometry).

Instead of finding a number, you rearrange to get one letter by itself.

Example 1: Solve for y

x + y = 10

We want y alone. Subtract x from both sides:

x - x + y = 10 - x
      y   = 10 - x

✅ Answer: y = 10 - x

Example 2: Solve for r (radius formula)

C = 2πr

We want r alone. Divide both sides by 2π:

C ÷ (2π) = 2πr ÷ (2π)
  C/(2π) = r

✅ Answer: r = C/(2π)

Example 3: Solve for h (volume of a box)

V = lwh

We want h alone. Divide both sides by lw:

V ÷ (lw) = lwh ÷ (lw)
  V/(lw) = h

✅ Answer: h = V/(lw)

💡 Pro Tip: Treat other letters like numbers! Just do the opposite operation to isolate the letter you want.

🎪 Summary: Your Equation-Solving Toolkit

graph TD
    A[Read the equation] --> B[Is x on both sides?]
    B -->|Yes| C[Move all x to one side]
    B -->|No| D[Start solving]
    C --> D
    D --> E[Undo + or - first]
    E --> F[Undo × or ÷ next]
    F --> G[Check your answer!]

The Golden Rules:

1. Balance is everything — What you do to one side, do to the other!
2. Work backwards — Undo operations in reverse order
3. Opposites attract — Use the opposite operation to undo
4. Always verify — Plug your answer back in to check

🌟 You Did It!

You now have the power to:

• ✓ Understand what equations are
• ✓ Verify solutions
• ✓ Solve one-step equations
• ✓ Tackle two-step equations
• ✓ Conquer multi-step equations
• ✓ Handle equations with fractions
• ✓ Rearrange literal equations

Remember: Every equation is just a puzzle waiting to be solved. You have the key—now go unlock those unknowns! 🔓