Solving Two-Step Equations

TL;DR

Two-step equations require you to isolate the variable by performing two inverse operations. Always tackle addition/subtraction first, then multiplication/division. The goal is to get the variable all by itself on one side of the equals sign.

1. The Mental Model

Think of an equation like a balanced scale. Whatever you do to one side, you must do to the other to keep it balanced. To "unwrap" the variable, you'll undo the operations in reverse order of how they were applied.

2. The Core Material

Solving a two-step equation means finding the value of the unknown variable. These equations involve two different operations (like multiplication and addition, or division and subtraction). To solve them, you'll use inverse operations to get the variable by itself.

The general strategy is:
1. Undo addition or subtraction: This means if you see + 5, you'll subtract 5 from both sides. If you see - 3, you'll add 3 to both sides.
2. Undo multiplication or division: If you see 2x, you'll divide both sides by 2. If you see x/4, you'll multiply both sides by 4.

You always work backward from the order of operations (PEMDAS/BODMAS). So, addition/subtraction are "undone" before multiplication/division.

graph TD
    A["Start with equation e.g., 2x + 5 = 11"] --> B{"Is there addition or subtraction?"}
    B -- Yes --> C["Undo addition/subtraction on both sides"]
    B -- No --> D{"Is there multiplication or division?"}
    C --> D
    D -- Yes --> E["Undo multiplication/division on both sides"]
    D -- No --> F["Variable is isolated!"]
    E --> F
    F --> G["Check your answer"]

Undoing Addition/Subtraction

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Let's say you have an equation like 3x + 7 = 16.
The + 7 is what you address first. To undo adding 7, you subtract 7.
3x + 7 - 7 = 16 - 7
This simplifies to 3x = 9.

Undoing Multiplication/Division

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Following from 3x = 9:
The 3x means 3 multiplied by x. To undo multiplying by 3, you divide by 3.
3x / 3 = 9 / 3
This simplifies to x = 3.

3. Worked Example

Let's solve the equation (x / 4) - 2 = 5.

1. Identify the first operation to undo: We see a subtraction of 2 (- 2). To undo this, we'll add 2 to both sides of the equation.
   (x / 4) - 2 + 2 = 5 + 2
   This simplifies to:
   x / 4 = 7

2. Identify the second operation to undo: Now we have x / 4. This means x is divided by 4. To undo this, we'll multiply both sides by 4.
   (x / 4) * 4 = 7 * 4
   This simplifies to:
   x = 28

3. Check your answer (optional but recommended!): Substitute x = 28 back into the original equation.
   (28 / 4) - 2 = 5
7 - 2 = 5
5 = 5
   Since both sides are equal, our answer x = 28 is correct!

4. Key Takeaways

• Always aim to get the variable completely by itself on one side of the equals sign.
• Use inverse operations: addition undoes subtraction, and multiplication undoes division.
• Undo addition/subtraction first, then multiplication/division (think "un-doing" PEMDAS backward).
• Whatever operation you perform on one side of the equation, you must perform on the other side.
• Checking your answer by plugging it back into the original equation is a great way to ensure accuracy.

Common Mistakes to Avoid

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• Performing multiplication/division before addition/subtraction.
• Only applying an operation to one side of the equation instead of both.
• Forgetting to distribute a negative sign if it's outside parentheses (though less common in basic two-step equations).
• Making arithmetic errors when adding, subtracting, multiplying, or dividing.

5. Now Try It

Solve the equation 6x - 10 = 32. Show each step clearly. Once you have your answer for x, plug it back into the original equation to verify your solution. Success looks like correctly isolating x and confirming your answer balances the equation.