$$ (16a + 4b + c) - (4a + 2b + c) = 360 - 120 \Rightarrow 12a + 2b = 240 \Rightarrow 6a + b = 120 \quad \text{(i)} $$
["Understanding the Equation: $$ (16a + 4b + c) - (4a + 2b + c) = 360 - 120 \Rightarrow 12a + 2b = 240 \Rightarrow 6a + b = 120 $$", "When solving linear equations, breaking them down step-by-step helps clarify the solution and strengthens algebraic rigor. One such algebraic expression involves simplifying the equation $$ (16a + 4b + c) - (4a + 2b + c) = 360 - 120 \Rightarrow 12a + 2b = 240 \Rightarrow 6a + b = 120 $$, a key stepping stone in many mathematical and real-world problem-solving scenarios.", "### The Original Equation Breakdown", "We begin with:\n$$(16a + 4b + c) - (4a + 2b + c) = 360 - 120$$", "On the left-hand side:\n- Expand the parentheses and simplify by combining like terms:\n$$\n(16a - 4a) + (4b - 2b) + (c - c) = 12a + 2b + 0 = 12a + 2b\n$$", "On the right-hand side:\n$$\n360 - 120 = 240\n$$", "Putting these together:\n$$\n12a + 2b = 240\n$$", "### Simplify to Standard Form", "Next, divide every term by 2 to simplify:\n$$\n\frac{12a}{2} + \frac{2b}{2} = \frac{240}{2} \Rightarrow 6a + b = 120\n$$", "### Final Result and Interpretation", "The simplified equation is:\n$$\n6a + b = 120\n$$", "This linear relationship between variables $a$ and $b$ opens the door to solving for one variable in terms of the other:\n$$\nb = 120 - 6a\n$$", "Such equations commonly appear in financing, optimization, and algebraic modeling—especially when balancing linear resources or costs.", "For example, if $a$ and $b$ represent quantities or values in a budget or production model, this equation ensures their weighted sum equals 120 after eliminating redundant terms like $c$. Understanding these transformations helps simplify complex expressions and improves accuracy in applied mathematics.", "---", "Key Takeaways:", "- Always simplify step-by-step for clarity and accuracy.\n- Eliminating like terms reduces complexity and uncovers core relationships.\n- Equations like $6a + b = 120$ form the basis for substitution and further problem-solving.", "By mastering these fundamental algebraic steps, anyone can confidently interpret and solve linear equations like those involving players $a$, $b$, and $c$ in expressions derived from real-world scenarios."]
