Perimeter: \(2(w + 3w) = 64\)

["Mastering the Equation: Solving Perimeter Problems Like (2(w + 3w) = 64)", "Understanding geometric formulas is essential for solving real-world problems, especially in areas like architecture, construction, and design. One commonly encountered equation is (2(w + 3w) = 64), which represents the perimeter of a rectangular shape where one dimension is (w) and another is thrice that width. In this article, we’ll walk through how to solve this algebraic equation step-by-step, explain its geometric meaning, and highlight practical applications of perimeter calculations.", "---", "### Understanding the Equation (2(w + 3w) = 64)", "At first glance, the expression (2(w + 3w)) may seem abstract, but it directly relates to the formula for the perimeter of a rectangle. Here’s what each term means:", "- (w) represents the width of the rectangle.\n- (3w) implies a length that is three times the width.\n- The parentheses ( (w + 3w) ) sum these two dimensions.\n- The factor 2 accounts for the perimeter being the total length around the shape — adding all four sides.", "Thus, (2(w + 3w) = 64) models a rectangle’s perimeter when length = 3 × width.", "---", "### Step-by-Step Solution", "Let’s solve the equation:", "[ \n2(w + 3w) = 64\n]", "1. Simplify inside the parentheses:\n [\n w + 3w = 4w\n ]\n So the equation becomes:\n [\n 2(4w) = 64\n ]", "2. Multiply:\n [\n 8w = 64\n ]", "3. Solve for (w):\n Divide both sides by 8:\n [\n w = \frac{64}{8} = 8\n ]", "Thus, the width (w) is 8 units.", "4. Find the length:\n Since length (= 3w),\n [\n \ ext{Length} = 3 \ imes 8 = 24\n ]", "---", "### Geometric Interpretation", "The solution confirms that for a rectangle with width 8 and length 24:", "- Perimeter = (2 \ imes (\ ext{length} + \ ext{width}) = 2 \ imes (24 + 8) = 2 \ imes 32 = 64)", "This matches the original equation, validating the algebra.", "---", "### Why This Equation Matters", "Equation-like problems such as (2(w + 3w) = 64) teach students to translate geometric shapes into algebraic expressions. Mastering this skill helps in various real-life applications:", "- Construction and Carpentry: Calculating wall or fencing perimeters.\n- Interior Design: Determining material needs for borders or trim.\n- Education: Developing algebraic reasoning through practical contexts.", "Understanding the link between algebra and geometry empowers learners to solve complex problems in mathematics, science, and engineering.", "---", "### Final Answer Recap", "[\n\boxed{w = 8} \quad \ ext{(Width)} \\n\boxed{\ ext{Length} = 24} \quad \ ext{(Three times the width)}\n]", "Perimeter equation (2(w + 3w) = 64) simplifies cleanly to (w = 8), revealing not just a number but a clear geometric configuration.", "---", "### Key Takeaways", "- Always simplify expressions inside parentheses before applying multiplication.\n- Relate geometric terms to algebraic variables for clarity.\n- Solve step-by-step using inverse operations.\n- Apply these principles to real-world perimeter and area problems.", "If you’re tackling similar algebraic equations, remember: geometry provides context, and algebra provides precision — together they unlock powerful problem-solving skills.", "---", "Keywords for SEO Optimization:\nPerimeter equation solution, solve (2(w + 3w) = 64), algebraic geometry, rectangle perimeter algebra, how to find width from perimeter, solve linear equations, perimeter formula application, step-by-step perimeter problem, real-world geometry problems", "---", "Use this guide whenever you need to solve or understand perimeter problems involving proportional dimensions — from homework to hands-on projects!"]