Solving Multi-Step Equations

TL;DR

Solving multi-step equations means isolating the variable by undoing operations in reverse order. Think of it like unwrapping a gift: the last thing you put on is the first thing you take off. Simplify each side first, then use inverse operations to get the variable by itself.

1. The Mental Model

Imagine your variable is a special item in a box. To get to it, you need to remove the layers surrounding it, one by one. You'll always remove the outermost layer first, which usually means addition/subtraction, then multiplication/division.

2. The Core Material

Multi-step equations are just like two-step equations, but with extra steps involved before you get to the familiar two steps. The goal is always the same: get the variable all by itself on one side of the equals sign.

Simplifying Each Side

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Before you start moving terms across the equals sign, make sure each side of the equation is as simple as possible. This involves two main things:

• Distribute: If you see parentheses with a number outside, multiply that number by everything inside the parentheses.
• Combine Like Terms: Look for terms on the same side of the equals sign that have the same variable and exponent (like 3x and 5x) or are just numbers (like 7 and 2). Combine them!

Using Inverse Operations

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Once each side is simplified, you'll start isolating the variable. You'll use inverse (opposite) operations to "undo" what's been done to the variable.

• To undo addition (+), use subtraction (-).
• To undo subtraction (-), use addition (+).
• To undo multiplication (*), use division (/).
• To undo division (/), use multiplication (*).

Remember the order: undo addition/subtraction first, then undo multiplication/division.

graph TD
    A["Start: Multi-step Equation"] --> B{"Any parentheses?"}
    B -- Yes --> C["Distribute any terms outside parentheses"]
    B -- No --> D{"Any like terms on the same side?"}
    C --> D
    D -- Yes --> E["Combine like terms on each side"]
    D -- No --> F["Undo addition or subtraction (move constants)"]
    E --> F
    F --> G["Undo multiplication or division (isolate variable)"]
    G --> H["Solution: Variable isolated"]

Dealing with Variables on Both Sides

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Sometimes, you'll see the variable on both sides of the equation (e.g., 3x + 5 = x - 7). Your first step after simplifying is to gather all the variable terms onto one side. It's usually easier to move the smaller variable term to avoid negative coefficients, but either way works.

For example, in 3x + 5 = x - 7, you could subtract x from both sides:
3x - x + 5 = x - x - 7
2x + 5 = -7

Now it looks like a standard two-step equation!

3. Worked Example

Let's solve: 4(2x - 3) + 6 = 2x + 18

1. Distribute: Apply the 4 to (2x - 3).
   8x - 12 + 6 = 2x + 18

2. Combine Like Terms on the left side: -12 + 6 becomes -6.
   8x - 6 = 2x + 18

3. Get variables on one side: Subtract 2x from both sides.
   8x - 2x - 6 = 2x - 2x + 18
6x - 6 = 18

4. Undo subtraction: Add 6 to both sides.
   6x - 6 + 6 = 18 + 6
6x = 24

5. Undo multiplication: Divide both sides by 6.
   6x / 6 = 24 / 6
x = 4

4. Key Takeaways

• Always simplify each side of the equation before moving terms across the equals sign.
• Use the distributive property first if you see parentheses.
• Combine like terms on the same side of the equation.
• To move terms across the equals sign, use inverse operations.
• Undo addition/subtraction before multiplication/division when isolating the variable.
• If variables are on both sides, move all variable terms to one side first.

Common Mistakes to Avoid:
- Forgetting to distribute a negative sign or distributing to only one term inside parentheses.
- Combining unlike terms (e.g., 3x + 5 does not become 8x).
- Performing an operation on one side but not the other side of the equation.
- Not simplifying each side fully before starting to isolate the variable.

5. Now Try It

Solve the equation 3(x + 5) - 7 = 2x - 4. Your success looks like finding the correct numerical value for x and being able to clearly show each step you took to get there, similar to the worked example.