The perimeter is \( 2(w + 3w) = 8w = 64 \).
["Understanding the Perimeter of a Rectangle: Solving ( 2(w + 3w) = 8w = 64 )", "When tackling geometry problems involving the perimeter of a rectangle, simplifying the expression correctly is key. One common equation you may encounter is:", "[\n2(w + 3w) = 8w = 64\n]", "This equation is derived from the basic formula for the perimeter of a rectangle, which is $ P = 2(\ ext{length} + \ ext{width}) $. In this case, the width is $ w $ and the length is $ 3w $, so substituting into the formula gives:", "[\nP = 2(w + 3w) = 2(4w) = 8w\n]", "Setting this equal to the known perimeter of 64 units:", "[\n8w = 64\n]", "To solve for $ w $, divide both sides by 8:", "[\nw = \frac{64}{8} = 8\n]", "So, the width is $ w = 8 $ units. Since the length is $ 3w $, that equals $ 3 \ imes 8 = 24 $ units.", "Why This Equation Matters in Geometry\nUnderstanding how perimeter formulas simplify helps students recognize relationships between dimensions and scale. By factoring and simplifying expressions like $ 2(w + 3w) = 8w $, learners develop algebraic thinking while reinforcing key geometric concepts.", "Practical Application\nKnowing the perimeter allows you to calculate how much fencing, border paint, or tape is needed around a rectangular area. With $ w = 8 $ and length $ 24 $, the total perimeter is exactly $ 64 $ units—making precise material estimates easy and effective.", "Summary\n- The perimeter formula starts with $ P = 2(\ ext{length} + \ ext{width}) $\n- Substituting $ w $ and $ 3w $ gives $ 2(w + 3w) = 8w $\n- Solving $ 8w = 64 $ yields $ w = 8 $\n- Length is $ 3w = 24 $, and the full perimeter is $ 64 $ units", "Mastering this algebraic manipulation builds a strong foundation for solving real-world measurement problems involving shapes and symmetry.", "---", "Keywords: perimeter, rectangle perimeter, solve 2(w + 3w) = 8w, algebraic geometry, how to solve perimeter equation, rectangular shape calculation, geometry problem solving", "Meta Description:\nLearn how to solve the equation ( 2(w + 3w) = 8w = 64 ) step by step. Discover how to use the rectangle perimeter formula ( P = 2(l + w) ), solve for width and length, and apply these skills to real-world measurement tasks."]
