Explore two different methodological approaches to solving a linear equation and understand their fundamental similarities | Step-by-Step Solution
Problem
Solve linear equation 3y + 12 + 2 - 2y = 6y + 3 - 3y through two different conceptual approaches
🎯 What You'll Learn
- Understand multiple approaches to solving linear equations
- Develop flexible algebraic reasoning skills
- Recognize different but equivalent problem-solving methods
Prerequisites: Basic algebraic manipulation, Equation solving techniques
💡 Quick Summary
I can see you're working with a linear equation that has multiple terms and variables on both sides - this is a great opportunity to explore how different solution strategies can lead to the same result! Here's something interesting to think about: when you look at both sides of your equation, what do you notice about the terms that are similar, and how might you group or rearrange them? You could try simplifying each side completely before moving terms around, or you could start moving like terms across the equals sign right away - both are perfectly valid approaches. What concepts do you remember about combining like terms and maintaining balance in an equation when you add or subtract the same thing from both sides? I'd encourage you to pick whichever method feels more natural to you first, then maybe try the other approach to see how they compare. You've got all the skills you need to tackle this - trust your understanding of the fundamentals!
Step-by-Step Explanation
What We're Solving:
We need to solve the linear equation 3y + 12 + 2 - 2y = 6y + 3 - 3y using two different conceptual approaches. This will help us see that there are multiple valid ways to tackle the same problem!The Approach:
Linear equations can be solved through different methods, but they all aim for the same goal: isolate the variable (y) on one side. I'll show you two approaches:- Approach 1: Simplify both sides first, then solve
- Approach 2: Move terms strategically without initial simplification
Step-by-Step Solution:
Approach 1: Simplify First Method
Step 1: Combine like terms on each side
- Left side: 3y + 12 + 2 - 2y = (3y - 2y) + (12 + 2) = y + 14
- Right side: 6y + 3 - 3y = (6y - 3y) + 3 = 3y + 3
- Our equation becomes: y + 14 = 3y + 3
- Subtract y from both sides: 14 = 2y + 3
- Subtract 3 from both sides: 11 = 2y
- Divide both sides by 2: y = 11/2 or y = 5.5
Approach 2: Strategic Moving Method
Step 1: Move all terms with y to the left side
- Original: 3y + 12 + 2 - 2y = 6y + 3 - 3y
- Subtract 6y from both sides: 3y + 12 + 2 - 2y - 6y = 3 - 3y
- Add 3y to both sides: 3y + 12 + 2 - 2y - 6y + 3y = 3
- Left side: (3y - 2y - 6y + 3y) + 12 + 2 = -2y + 14
- So we have: -2y + 14 = 3
- Subtract 14 from both sides: -2y = -11
- Divide both sides by -2: y = -11/(-2) = 11/2 or y = 5.5
The Answer:
Both approaches give us y = 5.5 (or y = 11/2)You can verify this by substituting back into the original equation - both sides will equal the same value!
Memory Tip:
Think of solving equations like organizing a messy room - you can either tidy up each area first (Approach 1) or move similar items together as you go (Approach 2). Both methods get you to a clean, organized result! The key is being systematic and keeping the equation balanced by doing the same operation to both sides.Great job working through this! Understanding multiple approaches makes you a more flexible and confident problem solver. 🌟
⚠️ Common Mistakes to Avoid
- Believing different solution methods mean different answers
- Not understanding the equivalence of algebraic transformations
- Losing track of algebraic steps during manipulation
This explanation was generated by AI. While we work hard to be accurate, mistakes can happen! Always double-check important answers with your teacher or textbook.
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📷 Problem detected:
Solve: 2x + 5 = 13
Step 1:
Subtract 5 from both sides...
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