The perimeter is \( 2(w + 2w + 3) = 54 \).
["# Solve the Perimeter Equation: ( 2(w + 2w + 3) = 54 )", "Understanding how to solve perimeter-related equations is essential in geometry and algebra. In this article, we’ll walk through step-by-step how to solve the equation ( 2(w + 2w + 3) = 54 ), explaining each step clearly and efficiently. Whether you’re a student learning algebra or a teacher seeking a solid example, this breakdown will help reinforce your mathematical skills.", "---", "## What is a Perimeter?", "The perimeter of a two-dimensional shape is the total distance around the boundary. For rectangles, the formula is:", "[\n\ ext{Perimeter} = 2(\ ext{length} + \ ext{width})\n]", "In our problem, the expression inside the parentheses combines length, a multiple of width, and a constant—making it a real-world inspired algebraic application.", "---", "## Step-by-Step Solution: ( 2(w + 2w + 3) = 54 )", "### Step 1: Simplify Inside the Parentheses", "Add the terms inside the parentheses:", "[\nw + 2w + 3 = 3w + 3\n]", "Now substitute back:", "[\n2(3w + 3) = 54\n]", "### Step 2: Expand the Expression", "Multiply 2 across the terms:", "[\n2 \cdot 3w + 2 \cdot 3 = 6w + 6\n]", "Now the equation is:", "[\n6w + 6 = 54\n]", "### Step 3: Isolate the Variable Term", "Subtract 6 from both sides to isolate the term with ( w ):", "[\n6w = 54 - 6\n]", "[\n6w = 48\n]", "### Step 4: Solve for ( w )", "Divide both sides by 6:", "[\nw = \frac{48}{6} = 8\n]", "---", "## Final Answer", "The width ( w ) of the shape is 8 units.", "Recall that the original expression ( w + 2w + 3 ) represents a real-world measurement where width ( w = 8 ) is combined with twice that width plus a fixed length of 3 units.", "---", "## Why This Equation Matters in Real-World Contexts", "Equations like ( 2(w + 2w + 3) = 54 ) often model physical situations—such as fencing a rectangular area or analyzing structural perimeters. Solving for ( w ) lets you determine exact dimensions needed for materials, cost estimation, or spatial planning.", "---", "## Summary of Key Calculations", "| Step | Calculation | Result |\n|-----------------------------|--------------------------------–|--------------|\n| Original expression | ( w + 2w + 3 ) | ( 3w + 3 ) |\n| Multiplied by 2 | ( 2(3w + 3) = 6w + 6 ) | |\n| Set equal to given value | ( 6w + 6 = 54 ) | |\n| Subtract 6 | ( 6w = 48 ) | |\n| Divide by 6 | ( w = 8 ) | ✅ Final Answer |", "---", "## Final Thoughts", "Solving ( 2(w + 2w + 3) = 54 ) demonstrates core algebraic techniques: simplifying expressions, expanding brackets, and isolating variables. Mastering these steps transforms abstract equations into practical problem-solving tools. Keep practicing to build confidence in working with geometric and algebraic models!", "---", "If you found this explanation helpful, share it to help others make sense of perimeter equations. Don’t forget to check related problems involving perimeters and algebraic modeling!", "---", "Keywords: perimeter equation, solve 2(w + 2w + 3) = 54, algebra problem, geometry perimeter, step-by-step solution, algebraic equation, algebra tutorial, perimeter word problem, solving linear equations, mathematical modeling."]
