Perimeter = 2(length + width) = 2(3x + x) = 8x = 64.
["# How to Solve Perimeter Problems: Understanding the Formula with Perimeter = 2(length + width) = 2(3x + x) = 8x = 64", "Understanding perimeter is essential in geometry, especially when calculating the boundaries of shapes like rectangles. Whether you're designing a garden, calculating materials for construction, or solving math problems, knowing how to work with the perimeter formula is key. In this article, we’ll break down how to find the value of a rectangle’s perimeter when the width is expressed in terms of a variable — specifically solving 2(3x + x) = 64 — and arrive at a clear expression for the length in terms of a single variable.", "---", "### What is Perimeter?", "Perimeter is the total distance around the outside of a polygon. For a rectangle, the perimeter formula is:", "[\n\ ext{Perimeter} = 2(\ ext{length} + \ ext{width})\n]", "This formula works because a rectangle has two equal lengths and two equal widths. Adding along all sides gives the total boundary.", "---", "### Applying the Formula: Given Length and Width Equations", "In many geometry problems, the length and width are given in algebraic expressions. For example, suppose the width is x and the length is 3x. Substitute these into the perimeter formula:", "[\n\ ext{Perimeter} = 2(\ ext{length} + \ ext{width}) = 2(3x + x)\n]", "Simplify the expression inside the parentheses:", "[\n3x + x = 4x\n]", "Now multiply by 2:", "[\n2(4x) = 8x\n]", "We’re told that this perimeter equals 64, so set up the equation:", "[\n8x = 64\n]", "---", "### Solving for x: Finding the Width", "Now solve for x, which represents the width in terms of the variable:", "[\nx = \frac{64}{8} = 8\n]", "So, the width is 8 units, and the length—being 3x—is:", "[\n3x = 3 \ imes 8 = 24\n]", "---", "### Final Verification", "To double-check, plug the values back into the perimeter formula:", "[\n\ ext{Perimeter} = 2(\ ext{length} + \ ext{width}) = 2(24 + 8) = 2 \ imes 32 = 64\n]", "This confirms the calculation is correct.", "---", "### Practical Takeaway", "When solving perimeter problems where dimensions are given algebraically (e.g., length = 3x, width = x), the formula becomes:", "[\n\ ext{Perimeter} = 2(3x + x) = 8x\n]", "Set (8x = 64) to find (x = 8), and determine the full dimensions from there. This systematic approach saves time and avoids common errors.", "---", "### Why This Matters in Real-World Applications", "Knowing how to derive perimeter equations helps in:", "- Interior design and flooring layout\n- Construction and landscaping (e.g., fencing, paving)\n- Mathematical problem-solving in STEM education", "By mastering algebra-based perimeter formulas, you gain a powerful tool for both geometry and practical applications.", "---", "### Summary", "- Perimeter of a rectangle: (P = 2(\ ext{length} + \ ext{width}))\n- When length = 3x and width = x, perimeter simplifies to (8x)\n- Given (8x = 64), solve to find (x = 8)\n- Width is 8 units, length is 24 units\n- Verification confirms accuracy and reinforces understanding", "Memorizing this step-by-step method supports confidence in solving similar algebra geometry problems every day.", "---", "# Key Benefits Recap:\n- Simplify perimeter expressions using variables\n- Apply basic arithmetic to solve linear equations\n- Verify answers by plugging back into original formulas\n- Apply knowledge to real-life measurements and planning", "Mastering these steps turns complex perimeter problems into simple, manageable steps."]
