$$ (36a + 6b + c) - (16a + 4b + c) = 720 - 360 \Rightarrow 20a + 2b = 360 \Rightarrow 10a + b = 180 \quad \text{(ii)} $$

["Understanding the Algebraic Equation: Solving $$ (36a + 6b + c) - (16a + 4b + c) = 720 - 360 \Rightarrow 20a + 2b = 360 \Rightarrow 10a + b = 180 $$", "Algebraic equations form the backbone of mathematical reasoning, essential in fields ranging from engineering to economics. One such equation—$$ (36a + 6b + c) - (16a + 4b + c) = 720 - 360 \Rightarrow 20a + 2b = 360 \Rightarrow 10a + b = 180 $$—illustrates how simplification and elimination lead to clear, actionable results. In this article, we break down step-by-step how to solve this equation, why it matters, and practical applications where expressions like $$ 10a + b = 180 $$ play key roles.", "---", "### Breaking Down the Equation: Step-by-Step Simplification", "We begin with the original expression:\n$$(36a + 6b + c) - (16a + 4b + c) = 720 - 360$$", "#### Step 1: Simplify the Left Side\nRemove parentheses carefully:\n$$36a + 6b + c - 16a - 4b - c$$\nNotice that $+c - c$ cancels out:\n$$(36a - 16a) + (6b - 4b) + (c - c) = 20a + 2b$$\nThus:\n$$20a + 2b = 720 - 360$$", "#### Step 2: Simplify the Right Side\nCalculate $720 - 360$:\n$$360$$\nSo:\n$$20a + 2b = 360$$", "#### Step 3: Reduce to Simplified Form\nDivide both sides by 2:\n$$\frac{20a + 2b}{2} = \frac{360}{2}$$\nWhich simplifies to:\n$$10a + b = 180$$\nThis is the final simplified line:\n$$10a + b = 180 \quad \ ext{(Equation (ii))}$$", "---", "### Why This Equation Matters: Real-World Applications", "Equations like $$10a + b = 180$$ are not just abstract—they model relationships between variables in various domains:", "- Business Finance: When $a$ and $b$ represent cost components (e.g., variable expenses per unit), this equation helps optimize pricing strategies. The constraint $10a + b = 180$ sets a budget cap for two interdependent factors.\n- Engineering Design: In resource allocation, where $a$ and $b$ symbolize materials or time units, simplifying expressions ensures efficient allocation under fixed capacity limits.\n- Data Science: Such linear relationships assist in regression models, where simplifying multiple variables leads to interpretable coefficients.", "---", "### How to Solve for Variables: Expressing Relationships Clearly", "To fully exploit this equation, solve for one variable in terms of the other:", "From\n$$10a + b = 180$$\nWe isolate $b$:\n$$b = 180 - 10a$$\nThis expresssion reveals an inverse relationship between $a$ and $b$: as $a$ increases, $b$ decreases proportionally to maintain the total of 180.", "---", "### Key Takeaways", "- Algebraic simplification—cancelling like terms and overcoming common terms—is powerful and accessible.\n- Starting with the original equation and simplifying stepwise ensures accuracy.\n- Each step serves a clear purpose: simplifying left-hand expression, reducing right-hand side, and reducing fractions.\n- Equations like $$10a + b = 180$$ bridge abstract math to real-world decision-making.", "---", "Final Equation Recap:\n$$20a + 2b = 360 \Rightarrow 10a + b = 180$$\nA foundational result, simple yet vital for modeling and solving practical problems involving two key variables.", "---", "Optimize Your Algebra Today: Whether you’re a student, educator, or professional, mastering equation solving empowers clearer thinking and effective problem-solving. Dive deeper into linear systems, and explore how structured simplification turns complexity into clarity.", "---", "By understanding how to reduce and solve equations like $$10a + b = 180$$, you unlock the ability to interpret and manipulate structured data across countless real-life scenarios—proving that the roots of algebra continue to nourish modern knowledge and innovation.", "---", "Keywords: algebra textbook, solving equations step-by-step, simplify algebraic expressions, $10a + b = 180$, linear equations explained, math problem solving, real-world algebra applications", "---", "For more insights into algebraic reasoning and equation simplification, visit our comprehensive algebra tutorials.", "---", "Note: The equation $$$20a + 2b = 360 \Rightarrow 10a + b = 180$$ is central to simplifying complex linear expressions, making algebra accessible for practical problem-solving across STEM fields. Understanding such steps builds intuition for advanced mathematics and scientific modeling."]