Chapter 14: Variables on Both Sides

Adapted from Wentworth's Elementary Algebra for MathDoctor.net

14.1 The "Gathering" Phase

Up until now, our variable x has lived on only one side of the equation. But what if we see 5x = 2x + 9? Before we can divide or subtract numbers, we must gather the variables onto one side.

Pro-Tip: Always move the smaller variable term. This helps you avoid dealing with negative numbers later in the problem!

14.2 The Step-by-Step Logic

Let's solve: 7x - 2 = 4x + 10

Step 1: Move the Variable. Subtract 4x from both sides to get all x's on the left.
7x - 4x - 2 = 10
3x - 2 = 10

Step 2: Move the Number. Add 2 to both sides.
3x = 12

Step 3: Isolate. Divide by 3.
x = 4

14.3 Special Cases: Infinite vs. No Solution

Sometimes, the variables completely cancel out. Don't panic! This tells you something special about the equation:

If you get something true (like 5 = 5), there are Infinite Solutions.
If you get something impossible (like 0 = 10), there is No Solution.

Chapter 14 Review

Solve for the variable:

8x = 5x + 15
3y - 7 = y + 1
10a + 2 = 4a + 20
2(x + 3) = 2x + 6
(Hint: Distribute first, then look at what's left!)

Chapter 14 Exercises

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Doctor's Recommendation: Moving Variables

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