$$ (10a + b) - (6a + b) = 180 - 120 \Rightarrow 4a = 60 \Rightarrow a = 15 $$
["### Solving $$ (10a + b) - (6a + b) = 180 - 120 $$ Step-by-Step Explanation", "Understanding how to solve algebraic equations is essential for mastering basic math and expanding your problem-solving skills. One frequently encountered equation in algebra—especially in word problems—is:", "$$\n(10a + b) - (6a + b) = 180 - 120\n$$", "This equation simplifies cleanly, revealing the value of key variables like $ a $. Let’s walk through the process step-by-step.", "#### Step 1: Simplify Both Sides of the Equation", "Start by simplifying the left-hand side:", "$$\n(10a + b) - (6a + b)\n$$", "Remove the parentheses, paying special attention to the minus sign before the second set:", "$$\n10a + b - 6a - b\n$$", "Now combine like terms:\n- For $ a $: $ 10a - 6a = 4a $\n- For $ b $: $ b - b = 0 $", "So the left side becomes:", "$$\n4a\n$$", "On the right-hand side:", "$$\n180 - 120 = 60\n$$", "Resulting in the simplified equation:", "$$\n4a = 60\n$$", "#### Step 2: Solve for $ a $", "Now isolate $ a $ by dividing both sides of the equation by 4:", "$$\na = \frac{60}{4} = 15\n$$", "#### Step 3: Why This Equation Matters", "Equations like this often appear in real-world scenarios—such as budgeting, physics, or financial calculations—where differences in linear expressions reveal meaningful values. Here, solving for $ a $ gives critical insight: when $ b $ cancels out, the equation effectively isolates the variable $ a $, simplifying otherwise complex expressions.", "If $ b $ represents an unknown quantity (like a hidden cost or variable input), subtracting it from both sides clearly shows $ a $'s direct link to the numeric difference (180 - 120 = 60).", "#### Final Thoughts", "Mastering simplification and linear equations equips you to tackle more advanced algebra confidently. Remember:\n- Distribute carefully\n- Combine like terms promptly\n- Isolate variables using inverse operations", "So, from $$ (10a + b) - (6a + b) = 180 - 120 $$, we clearly prove:", "$$\n4a = 60 \Rightarrow a = 15\n$$", "This straightforward algebra not only solves the equation—but builds the foundation for success in standardized tests, STEM fields, and everyday problem-solving.", "---", "Keywords: Algebra equation solution, solving for $ a $, linear equations, step-by-step algebra, simplify expressions, $ 4a = 60 $, $ a = 15 $, canceling variables, arithmetic simplification, beginner algebra.", "Use this guide to strengthen your algebraic fluency and unlock clearer reasoning in future math challenges!"]
