The perimeter is \( 2(w + 3w) = 64 \).

["# Solving the Perimeter Equation: ( 2(w + 3w) = 64 )", "Understanding basic geometry is essential for solving practical problems, and one common challenge is working with the perimeter of rectangles. Today, we’ll explore a key algebraic problem involving perimeter: solving ( 2(w + 3w) = 64 ). This equation helps not only find the width of a rectangle but also reinforces fundamental concepts in algebra and geometry.", "## Understanding the Perimeter Formula", "The perimeter of a rectangle is calculated using the formula:\n[\n\ ext{Perimeter} = 2 \ imes (\ ext{length} + \ ext{width})\n]\nIn many problems, the width ( w ) is known or related to another dimension. Here, the expression inside the parentheses ( w + 3w ) indicates that the width ( w ) is repeated three times — reflecting the even sides of a rectangle composed of segments related by a factor of 3.", "Given:\n[\n2(w + 3w) = 64\n]", "## Step-by-Step Solution", "### Step 1: Simplify the expression inside the parentheses\nCombine like terms:\n[\nw + 3w = 4w\n]\nSo the equation becomes:\n[\n2(4w) = 64\n]", "### Step 2: Multiply inside the parentheses\n[\n8w = 64\n]", "### Step 3: Solve for ( w )\nDivide both sides by 8:\n[\nw = \frac{64}{8} = 8\n]", "## The Solution", "The width of the rectangle is ( w = 8 ). Since ( 3w = 3 \ imes 8 = 24 ), the length of the rectangle is 24 units.", "Verifying the perimeter:\n[\n2(w + 3w) = 2(8 + 24) = 2 \ imes 32 = 64\n]\nThe calculation confirms the solution.", "## Why This Equation Matters", "Solving ( 2(w + 3w) = 64 ) is more than just algebra—it’s a real-world modeling exercise. Such problems are common in architecture, construction, and design, where precise measurements ensure structural integrity and efficiency. Understanding how to interpret and solve these equations enables better spatial reasoning and practical problem-solving skills.", "## Apply the Concept", "To find ( w ) in any similar perimeter problem, focus on:\n- Combining like terms in parentheses\n- Multiplying coefficients correctly\n- Isolating the variable step-by-step", "This core approach applies to countless geometry and algebra scenarios beyond simple rectangle perimeters.", "---", "Key takeaways:\n- Perimeter of rectangle: ( 2(\ ext{length} + \ ext{width}) )\n- Simplify expressions before solving\n- Always verify your solution", "Master these techniques to confidently tackle perimeter problems and strengthen your math foundation."]