Lesson 2.7
Solving with Like Terms
Before you start moving things across the equal sign, you need to clean up the mess. Combining like terms simplifies the equation into a friendly two-step problem.
Introduction
Imagine trying to organize a room. You wouldn't move one sock to the drawer, then one shirt, then another sock. You'd pile all the socks together first. Solving equations is the same: simplify each side before moving terms.
Past Knowledge
You can combine like terms (e.g., ).
Today's Goal
Combine terms on the same side of the equals sign to create a two-step equation.
Future Success
This is the first step in the "General Strategy" for solving any linear equation.
Key Concepts
The "Wall" Rule
The equal sign is a wall.
If terms are on the SAME side, combine them (Addition/Subtraction).
If terms are on OPPOSITE sides, use inverse operations to move them.
Steps to Solve
- Identify Like Terms: Look for variables with the same exponent or constant numbers on the same side.
- Combine: Add or subtract their coefficients. Keep the sign with the number!
- Solve: You now have a simple two-step equation. Undo addition, then multiplication.
Worked Examples
Example 1: Combining Variables
BasicSolve for :
Combine Like Terms
and are on the same side.
Solve Two-Step Equation
Example 2: Combining Constants
IntermediateSolve for :
Combine Constants
Combine and .
Solve Two-Step Equation
Example 3: Tricky Signs
AdvancedSolve for :
Combine Like Terms (Right Side)
Combine and .
Solve (Add 9, then Divide)
Common Pitfalls
Opposite Operations on Same Side
For , students sometimes subtract from . Incorrect! Since they are on the same side, just add them: .
Combining Unlike Terms
You cannot combine and to get . One has an , the other doesn't. They are apples and oranges.
Real-Life Applications
Inventory management relies on this. If a store has shirts in the back, adds more from a shipment, and puts 5 on display:
Simplifying to makes it much easier to track stock levels quickly.
Practice Quiz
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