The perimeter is \( 2(x + 3x) = 8x = 64 \).

["Understanding the Perimeter Formula: Solving ( 2(x + 3x) = 8x = 64 ) Step-by-Step", "When learning geometry, one of the most fundamental concepts is calculating the perimeter of a figure. The perimeter is the total distance around the boundary, and for many regular shapes like rectangles, it involves adding together the lengths of all sides. A common algebraic problem that appears in equations resembling ( 2(x + 3x) = 8x = 64 ) helps students practice simplifying expressions and solving for unknown variables.", "In this article, we’ll break down the expression ( 2(x + 3x) = 64 ), solve for the unknown variable ( x ), and explore how this connects to real-world geometry applications.", "---", "### Step 1: Simplify the Perimeter Expression", "The perimeter of a rectangle, for example, is calculated as:", "[\nP = 2 \ imes (\ ext{length} + \ ext{width})\n]", "In algebra, when the length and width involve a common factor, expressions simplify neatly. Here, we see:", "[\nx + 3x = 4x\n]", "So the expression becomes:", "[\n2(x + 3x) = 2(4x) = 8x\n]", "This matches the left side in the given equation.", "---", "### Step 2: Set Up the Equation", "The problem states:", "[\n2(x + 3x) = 64\n]", "We now substitute the simplified form:", "[\n8x = 64\n]", "---", "### Step 3: Solve for ( x )", "To isolate ( x ), divide both sides of the equation by 8:", "[\nx = \frac{64}{8} = 8\n]", "So, the value of ( x = 8 ).", "---", "### Step 4: Calculate the Final Perimeter (Bonus Insight)", "Though not always necessary, verifying satisfies the original problem:", "- If ( x = 8 ), then:\n - Width = ( x = 8 )\n - Length = ( 3x = 3 \ imes 8 = 24 )", "- Perimeter = ( 2(\ ext{length} + \ ext{width}) = 2(24 + 8) = 2 \ imes 32 = 64 ), which matches the given perimeter.", "---", "### Real-World Use of This Problem", "This type of perimeter equation is essential in architecture, construction, and interior design where precise boundary measurements are crucial. Knowing how to translate geometric formulas into algebraic equations helps professionals solve for unknown dimensions quickly. Similarly, for students, mastering such problems builds a strong foundation in algebra and geometry.", "---", "### Summary", "- The expression ( 2(x + 3x) ) simplifies to ( 8x ).\n- Solving ( 8x = 64 ) yields ( x = 8 ).\n- This validates the original perimeter equation geometrically.\n- Applications include any real-world scenario requiring precise boundary calculations.", "---", "Keywords: perimeter formula, solve for x, algebraic perimeter, geometry algebra, 2(x + 3x) = 8x = 64, solving equations, rectangle perimeter, math tutorial, algebra practice, step-by-step perimeter", "---", "If you found this guide helpful, share it with classmates or bookmark it for future geometry studies — understanding perimeter and algebra is key to mastering basic math and real-world problem-solving!"]