\( 3x + 5(20 - x) = 80 \)

["# Solve the Equation: ( 3x + 5(20 - x) = 80 )", "Solving linear equations is a fundamental skill in algebra, and equations like ( 3x + 5(20 - x) = 80 ) appear frequently in math problems and classroom learning. Whether you're preparing for exams or developing problem-solving skills, understanding how to simplify and solve equations like this one is essential. In this article, we’ll break down step-by-step how to solve ( 3x + 5(20 - x) = 80 ), explore the reasoning behind each step, and highlight its relevance in real-world math applications.", "## Understanding the Equation", "The equation ( 3x + 5(20 - x) = 80 ) combines distributive properties, combining like terms, and isolating variables—core components of algebraic problem-solving. On the surface, it may look complex, but by methodically expanding and simplifying, the solution becomes clear and manageable.", "## Step-by-Step Solution", "### Step 1: Distribute the 5", "Start by eliminating the parentheses using the distributive property:\n[\n3x + 5(20 - x) = 80 \implies 3x + 100 - 5x = 80\n]", "### Step 2: Combine Like Terms", "Now combine the ( x )-terms on the left-hand side:\n[\n(3x - 5x) + 100 = 80 \implies -2x + 100 = 80\n]", "### Step 3: Isolate the Variable", "Subtract 100 from both sides:\n[\n-2x = 80 - 100 \implies -2x = -20\n]\nNow divide both sides by (-2):\n[\nx = \frac{-20}{-2} = 10\n]", "## Verifying the Solution", "To confirm correctness, substitute ( x = 10 ) back into the original equation:\n[\n3(10) + 5(20 - 10) = 30 + 5(10) = 30 + 50 = 80\n]\nThe left-hand side equals the right-hand side, confirming that the solution is accurate.", "## Real-Life Applications", "Equations like ( 3x + 5(20 - x) = 80 ) model real-world scenarios such as budgeting, mixing solutions, or balancing resources. For instance, suppose ( x ) represents the number of units of one product, and ( 20 - x ) another, with associated costs or quantities. Solving such equations helps optimize decisions and understand trade-offs in everyday problems.", "## Conclusion", "Mastering equation-solving techniques such as distributing terms, combining like terms, and isolating variables is key to success in algebra. Understanding ( 3x + 5(20 - x) = 80 ) not only strengthens your math foundation but also equips you to tackle practical challenges with confidence. Keep practicing—consistent application of these steps builds fluency and intuition in algebra.", "---", "Keywords: solve linear equation, algebraic equation, 3x + 5(20 - x) = 80, step-by-step algebra, equation solving, distributive property, algebraic reasoning, middle school math, high school algebra."]