Perimeter: \( 2(w + 2w + 3) = 54 \).

["Understanding the Perimeter Equation: ( 2(w + 2w + 3) = 54 ) – Step-by-Step Guide", "When solving geometric problems involving perimeters, equations often model real-world situations. One such essential equation is:", "[\n2(w + 2w + 3) = 54\n]", "This formula helps find the value of ( w ), which commonly represents a side length in a geometric figure—such as a rectangular garden or rectangular frame—where perimeter calculations are key.", "---", "### What Is Perimeter and Why Does It Matter?", "Perimeter is the total distance around a two-dimensional shape. For rectangles, the perimeter ( P ) is calculated as:\n[\nP = 2 \ imes (\ ext{length} + \ ext{width})\n]", "In this equation, the expression inside the parentheses (( w + 2w + 3 )) models the individual sides and an added dimension, and multiplying by 2 accounts for all four sides (two lengths and two widths).", "---", "### Step-by-Step Solution: Solving ( 2(w + 2w + 3) = 54 )", "1. Simplify the expression inside the parentheses:\n Combine like terms:\n [\n w + 2w + 3 = 3w + 3\n ]", "So the equation becomes:\n [\n 2(3w + 3) = 54\n ]", "2. Distribute the 2:\n [\n 6w + 6 = 54\n ]", "3. Subtract 6 from both sides:\n [\n 6w = 54 - 6 = 48\n ]", "4. Divide by 6:\n [\n w = \frac{48}{6} = 8\n ]", "---", "### Final Answer", "The value of ( w ) is ( \boxed{8} ). This means each unit-length dimension corresponds to 8 units. In practical terms, this could represent a side of a rectangle where the total perimeter is 54 units.", "---", "### Real-World Application Example", "Imagine building a garden with rectangular shape where:\n- One side is ( w = 8 ) feet,\n- The adjacent side (twice ( w )) is ( 2w = 16 ) feet,\n- There’s an added border of width 3 feet along each side.", "Then the full perimeter calculates as:\n[\n2(8 + 16 + 3) = 2(27) = 54 \ ext{ feet}\n]", "Thus, the equation ( 2(w + 2w + 3) = 54 ) accurately reflects this scenario.", "---", "### Summary", "- The perimeter equation ( 2(w + 2w + 3) = 54 ) models a simplified rectangular configuration.\n- Solving it step-by-step yields ( w = 8 ).\n- Such equations are foundational for studying geometry, architecture, and real-world measurements.", "Mastering these algebraic techniques strengthens problem-solving skills essential in math, physics, and engineering fields. Whether for homework, exams, or practical design, understanding this perimeter equation is a powerful tool.", "---", "Keywords: perimeter equation, solve 2(w + 2w + 3) = 54, geometric problems, algebra practice, rectangular perimeter, step-by-step solving, w = 8, real-world math."]