Perimeter: \(2(w + 4w) = 90\).

["# Solving the Perimeter Equation: (2(w + 4w) = 90)", "Understanding perimeter is essential in geometry, especially when working with shapes like rectangles. One common problem students encounter is solving perimeter equations such as (2(w + 4w) = 90). This article explains how to solve this equation step-by-step, highlights the mathematical principles behind it, and provides practical insight into finding the width (w) when given the total perimeter.", "## What is the Perimeter?", "The perimeter of a rectangle is the total distance around its edges. For a rectangle with width (w) and length (l), the formula is:", "[\n\ ext{Perimeter} = 2 \ imes (\ ext{Length} + \ ext{Width}) = 2(l + w)\n]", "When the length is expressed in terms of the width—such as (l = 4w)—this leads to expressions like (2(w + 4w)).", "## Simplify the Equation", "Start with the given equation:", "[\n2(w + 4w) = 90\n]", "First, combine like terms inside the parentheses:", "[\nw + 4w = 5w\n]", "So the equation becomes:", "[\n2(5w) = 90\n]", "Multiply:", "[\n10w = 90\n]", "## Solve for (w)", "Now isolate (w) by dividing both sides by 10:", "[\nw = \frac{90}{10} = 9\n]", "Thus, the width (w) is 9 units.", "## Find the Length", "Since the length is four times the width:", "[\nl = 4w = 4 \ imes 9 = 36\n]", "## Verify the Solution", "Plug (w = 9) and (l = 36) back into the original perimeter formula:", "[\n2(l + w) = 2(36 + 9) = 2 \ imes 45 = 90\n]", "The calculated perimeter matches the given value, confirming our solution is correct.", "## Why This Equation Matters", "Solving perimeter equations is more than just algebra practice—it builds foundational problem-solving skills used in architecture, construction, and design. Knowing how to manipulate and simplify expressions like (2(w + 4w)) helps students understand real-world spatial reasoning.", "## Tips for Solving Similar Problems", "- Always simplify expressions before solving.\n- Remember that perimeter uses the sum of width and length doubled.\n- Express all variables clearly using given relationships.\n- Always check your solution by substituting back into the original equation.", "## Conclusion", "The perimeter equation (2(w + 4w) = 90) neatly demonstrates how algebraic manipulation helps uncover unknown dimensions of a rectangle. By simplifying and solving step-by-step, you find (w = 9), a practical skill applicable across math and real-life scenarios. Mastering these steps boosts confidence in geometry and beyond.", "---", "Key Terms: perimeter equation, solving (2(w + 4w) = 90), rectangle dimensions, algebra for geometry.\nSEO Keywords: solve (2(w + 4w)), perimeter formula, rectangle width calculation, algebraic perimeter problems."]