Lesson 2.4

Two-Step Equations

When dressing, you put on socks then shoes. To take them off, you reverse the order: shoes first, then socks. Solving equations works the same way—we undo operations in reverse order.

Introduction

A two-step equation involves two operations (like multiplication and addition). To solve it, we must peel away the layers surrounding the variable. The rule is simple: Reverse the Order of Operations.

Past Knowledge

You simplify expressions using GEMDAS (Groups, Exponents, Mult/Div, Add/Sub).

Today's Goal

Solve equations by undoing Addition/Subtraction first, then Multiplication/Division.

Future Success

This "Reverse GEMDAS" strategy works for almost every equation you will encounter in math.

Key Concepts

The "SADMEP" Strategy

To build an expression, we use GEMDAS.
To solve (destroy) an equation, we go backwards: SADMEP.

Step 1

Undo (+/−)

Remove the Constant

→
Step 2

Undo (×/÷)

Remove the Coefficient

Visualizing the Layers

Think of the variable as the prize inside two boxes.

  • Outer Box (Constant): The number added or subtracted (e.g., $+5$). Take this off first.
  • Inner Box (Coefficient): The number multiplying the variable (e.g., $2x$). Take this off second.

Worked Examples

Example 1: The Standard Two-Step

Basic

Solve for :

1

Step 1: Undo Addition

Subtract 5 from both sides to remove the constant.

==
2

Step 2: Undo Multiplication

Divide by 2 to isolate .

Example 2: Division First? No!

Intermediate

Solve for :

1

Step 1: Undo Subtraction

Add 4 to both sides.

==
2

Step 2: Undo Division

Multiply by 3.

Example 3: Tricky Formatting

Advanced

Solve for :

Watch Out: Is the 6 positive or negative? It's positive (no sign in front). The negative sign belongs to the 3x.
1

Step 1: Remove the Constant

Subtract 6 (not add 6) to remove the positive 6.

==
2

Step 2: Divide by Negative

Divide by -3 (don't lose the sign!).

Common Pitfalls

Dividing Too Soon

In , if you divide by 2 first, you must divide everything (including the 5). It creates messy fractions. Always subtract/add first!

The "Hanging Negative"

In , after subtracting 10, students often drop the negative on the $2x$. Remember: the sign to the left stays with the term. It becomes .

Real-Life Applications

Most real-life fees are two-step equations. A taxi might charge a $5 flat fee plus $2 per mile. If your ride cost $15, you use a two-step equation to find the miles:

  1. Subtract the flat fee ($15 - 5 = 10$)
  2. Divide by the rate ($10 \div 2 = 5$ miles)

Practice Quiz

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