How to Solve: Variables on Both Sides

How to Solve: Variables on Both Sides

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"If you can solve equations with variables on both sides, you can crack word problems about phone plans, sports scores, and even college-level physics—this is the skill that separates ‘I kinda get it’ from ‘I own this exam.’"

What You Need To Know First

Before tackling variables on both sides, you must already understand: 1. Solving one-step and two-step equations (e.g., 3x + 5 = 14 or 2x – 7 = 3). 2. Combining like terms (e.g., 4x + 2x = 6x). 3. Distributive property (e.g., 3(x + 2) = 3x + 6).

If any of these feel shaky, pause here and review them first.

Key Vocabulary

Formulas To Know

(No formulas to memorize—just methods! But here’s what you’ll use:)

1. Addition/Subtraction Property of Equality
2. If a = b, then a + c = b + c and a – c = b – c.

3. Use: Move variables or constants to one side.

4. Multiplication/Division Property of Equality

5. If a = b, then a × c = b × c (if c ≠ 0) and a ÷ c = b ÷ c (if c ≠ 0).

6. Use: Isolate the variable after combining like terms.

7. Distributive Property

8. a(b + c) = ab + ac.
9. Use: Expand parentheses before solving.

Step-by-Step Method

Goal: Get all variables on one side and constants on the other.

1. Distribute (if there are parentheses).
2. Combine like terms on each side separately.
3. Move variables to one side using inverse operations (add/subtract).
4. Move constants to the other side using inverse operations.
5. Isolate the variable (divide or multiply).
6. Check your answer by plugging it back into the original equation.

Worked Example Using the Steps

Equation: 3x + 5 = 2x + 10

Worked Examples

Example 1 – Basic

Equation: 4x – 3 = 2x + 7

What we did and why: We moved 2x to the left to get all variables on one side, then moved constants to isolate x. Always check your answer!

Example 2 – Medium (Parentheses + Fractions)

Equation: 2(x + 4) = 3x – 5

What we did and why: We expanded the parentheses first, then moved variables to one side. Fractions or decimals? Clear them early (not needed here).

Example 3 – Exam Style (Disguised Variables)

Problem: "The sum of a number and 8 is equal to three times the number minus 4. Find the number."

What we did and why: We translated the word problem into an equation, then solved it like any other. Always define your variable first!

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, listen up—this is your 60-second crash course for variables on both sides. Here’s the deal:

1. Distribute first if you see parentheses. No shortcuts.
2. Get all x’s on one side by adding or subtracting. Pick a side and stick with it.
3. Move constants to the other side—same rule: add or subtract.
4. Divide or multiply to isolate x.
5. Check your answer by plugging it back in. If it doesn’t work, you messed up—go back!

Examiners love to hide parentheses or throw in word problems. Stay calm, define your variable, and follow the steps. You’ve got this. Now go practice—tonight’s the night you master this!