Perimeter: \( 2(w + 3w) = 400 \)

["# Solving the Linear Equation: Perimeter of a Rectangle with ( 2(w + 3w) = 400 )", "Understanding how to calculate the perimeter of a rectangle is fundamental in geometry, and solving linear equations helps sharpen algebraic thinking. In this article, we’ll walk through solving the equation ( 2(w + 3w) = 400 ), explaining each step clearly. This method demonstrates both algebraic manipulation and real-world application—perfect for students, teachers, or anyone curious about geometry and equations.", "## Understanding Rectangle Perimeter", "Before diving into the calculation, let’s recall the formula for the perimeter ( P ) of a rectangle:\n[ P = 2(\ ext{length} + \ ext{width}) ]", "If we let the width be ( w ) and the length is given as ( 3w ), the perimeter becomes:\n[ P = 2(w + 3w) ]", "## Step-by-Step Solution to ( 2(w + 3w) = 400 )", "### Step 1: Simplify the Expression Inside the Parentheses\nStart by combining like terms inside the parentheses:\n[ w + 3w = 4w ]\nSo the equation becomes:\n[ 2(4w) = 400 ]", "### Step 2: Multiply\nMultiply the constants:\n[ 2 \ imes 4w = 8w ]\nNow the equation is:\n[ 8w = 400 ]", "### Step 3: Solve for ( w )\nTo isolate ( w ), divide both sides by 8:\n[ w = \frac{400}{8} ]\n[ w = 50 ]", "### Step 4: Find the Length\nSince the length is ( 3w ):\n[ \ ext{Length} = 3 \ imes 50 = 150 ]", "### Step 5: Verify the Perimeter\nDouble-check by plugging the values back in:\n[ \ ext{Perimeter} = 2(150 + 50) = 2 \ imes 200 = 400 ]\nThis confirms the solution.", "## Real-World Application", "This type of problem appears in architecture, construction, and design, where calculating material needs requires knowing dimensions from perimeter constraints. Understanding how to translate word problems into equations—and solve them—is essential.", "## Final Answer", "The width ( w ) is 50 units, and the length is 150 units. The equation ( 2(w + 3w) = 400 ) correctly models this rectangle’s perimeter and solves to ( w = 50 ).", "---", "### Key Takeaways", "- Simplify inside parentheses: Always combine like terms first.\n- Multiplication inside parentheses: Apply distributive property carefully.\n- Isolate the variable: Use basic algebraic operations to solve.\n- Verify your work: Replacing values ensures correctness.", "Mastering such equations strengthens your ability to tackle more complex geometry and algebra problems, bringing clarity and confidence to mathematical problem solving.", "---", "If you're studying geometry or algebra, practicing equations like ( 2(w + 3w) = 400 ) builds a solid foundation. Remember: every parameter has a real meaning—in mastering math, you master clarity."]