["Mastering the Equation: Solve $12a + 8(15 - a) = 156$ Step-by-Step", "Solving linear equations is a fundamental skill in algebra, crucial for students, educators, and anyone tackling real-world problems involving variables. Today, we break down how to solve the equation $$12a + 8(15 - a) = 156$$—a clear and practical example that highlights key algebraic techniques. Whether you're preparing for exams or looking to sharpen your math skills, this guide will walk you through the solution step-by-step.", "---", "### Understanding the Equation", "The equation $$12a + 8(15 - a) = 156$$ contains a variable $$a$$ mixed with arithmetic operations. Our goal is to isolate $$a$$ and find its exact value. Most importantly, solving this equation demonstrates how to handle distribution, combine like terms, and simplify complex expressions—skills widely applicable in science, engineering, and daily problem-solving.", "---", "### Step 1: Expand the Parentheses", "Start by applying the distributive property:
\n$$8(15 - a) = 8 \cdot 15 - 8 \cdot a = 120 - 8a.$$", "Now substitute back into the original equation:
\n$$12a + (120 - 8a) = 156.$$", "---", "### Step 2: Combine Like Terms", "Rewrite the equation without parentheses and combine terms involving $$a$$:
\n$$12a - 8a + 120 = 156$$
\n$$\Rightarrow 4a + 120 = 156.$$", "This simplifies our work by reducing the number of terms and making isolation of $$a$$ straightforward.", "---", "### Step 3: Isolate the Variable", "Subtract 120 from both sides to eliminate the constant:
\n$$4a = 156 - 120$$
\n$$4a = 36.$$", "Now divide both sides by 4:
\n$$a = \frac{36}{4} = 9.$$", "---", "### Step 4: Verify the Solution", "Plug $$a = 9$$ back into the original equation to confirm correctness:
\n$$12(9) + 8(15 - 9) = 108 + 8(6) = 108 + 48 = 156.$$
\nThe left side equals the right side, validating our solution.", "---", "### Why This Equation Matters", "- Educational Value: This problem teaches essential algebra skills: distribution, combining terms, and solving linear equations.
\n- Real-World Application: Similar equations appear in budgeting, physics (e.g., motion and force calculations), and computer programming.
\n- Confidence Build: Mastering this equation strengthens algebraic intuition, preparing learners for more advanced math topics like systems of equations and linear modeling.", "---", "### Summary", "To solve $$12a + 8(15 - a) = 156$$, follow these steps:
\n1. Expand using the distributive property.
\n2. Combine like terms.
\n3. Isolate the variable.
\n4. Solve and verify.", "With $$a = 9$$, you’ve successfully transformed a complex expression into a clear solution—showcasing the power and elegance of algebraic problem-solving.", "---", "Keywords: linear equation solving, algebra techniques, solve 12a + 8(15 - a) = 156, step-by-step algebra, equation solution method, educational algebra, real-world math applications, distribute and combine like terms, verify solution."]
