\(2(w + w + 150) = 1400\)

["# Solving the Equation: (2(w + w + 150) = 1400)", "When solving math problems like (2(w + w + 150) = 1400), it's essential to break the equation down step by step to uncover the value of (w). This algebraic equation appears in many real-world scenarios, such as budgeting, sales calculations, and distance-rate-time problems. Understanding how to solve it not only helps with algebra proficiency but also strengthens logical thinking.", "## Step-by-Step Solution", "### Step 1: Simplify inside the Parentheses\nStart by simplifying the expression within the parentheses:\n[\nw + w + 150 = 2w + 150\n]\nSo the equation becomes:\n[\n2(2w + 150) = 1400\n]", "### Step 2: Distribute the 2\nMultiply the 2 across the parentheses:\n[\n2 \ imes 2w = 4w \quad \ ext{and} \quad 2 \ imes 150 = 300\n]\nNow the equation is:\n[\n4w + 300 = 1400\n]", "### Step 3: Isolate the Variable\nSubtract 300 from both sides to eliminate the constant:\n[\n4w = 1400 - 300 = 1100\n]", "### Step 4: Solve for (w)\nDivide both sides by 4 to find (w):\n[\nw = \frac{1100}{4} = 275\n]", "---", "## Final Answer", "[\nw = 275\n]", "### What does this mean?\nIf (w) represents a quantity—such as price, time, or units—plugging (w = 275) back into the original equation confirms correctness:\n[\n2(275 + 275 + 150) = 2(700) = 1400\n]\nThis validates our solution.", "---", "## Practical Applications of This Equation", "Equations like (2(w + w + 150) = 1400) often model situations involving repeated costs or combined measurements. For example:", "- Budgeting & Finances: If (w) represents a base cost repeated twice with a fixed overhead, this models total spending over two identical items plus expenses.\n- Education & Testing: Such equations appear in problems requiring similar values to meet a total target.\n- Science & Engineering: Repeated measurements with constant values can be modeled algebraically for problem-solving.", "---", "## Why Learn to Solve Such Equations?", "Understanding how to solve equations trains your ability to analyze relationships between quantities. It builds a foundation for more advanced math, including systems of equations, linear programming, and real-world data interpretation.", "---", "### Key Takeaways", "- Always simplify expressions carefully.\n- Perform operations systematically: distribute, combine like terms, isolate variables.\n- Verify solutions by substituting back into the original equation.\n- Recognize real-life contexts where such equations model problems.", "Mastering equations like (2(w + w + 150) = 1400) unlocks greater confidence in algebra and enhances logical reasoning useful across academic and practical domains.", "---", "Keywords:\nequation solving, algebra problems, step-by-step tutorial, solving linear equations, (2(w + w + 150) = 1400), mathematical reasoning, real-world math applications, equation simplification, intermediate algebra."]