Substitute the expressions for length and width: \( 2(3w + w) = 64 \).

["Title: Simplify and Solve: Substituting Expressions for Length and Width in Perimeter Problems", "When solving geometry problems involving the perimeter of a rectangle, one of the most effective strategies is substituting real-world measurements into algebraic expressions. Consider the equation ( 2(3w + w) = 64 ), which appears frequently when calculating the perimeter of a rectangular shape where side lengths are expressed in terms of a variable.", "### Understanding the Expression", "The expression ( 2(3w + w) ) models the perimeter of a rectangle, not with separate values for length and width, but by expressing both in terms of a single variable, ( w )—this could represent the width or any fixed dimension. Breaking it down:", "- ( w ) represents the width.\n- ( 3w ) represents the length.\n- The sum ( 3w + w = 4w ) combines the total of adjacent sides.\n- Multiplying by 2 gives the full perimeter: ( 2(3w + w) = 2 \ imes 4w = 8w ).", "So, the original equation simplifies directly to:", "[\n8w = 64\n]", "This step highlights how substitution replaces tangible dimensions with variables, transforming practical measurements into algebraic expressions for easier computation.", "### Solving the Equation", "Now we solve:", "[\n8w = 64\n]", "Divide both sides by 8:", "[\nw = \frac{64}{8} = 8\n]", "Thus, the width is ( w = 8 ). To find the length, recall ( \ ext{length} = 3w = 3 \ imes 8 = 24 ).", "### Why This Substitution Works", "Replacing specific values for length and width with algebraic expressions streamlines problem-solving:", "- It turns word-based problems into solvable equations.\n- It reduces ambiguity by clearly linking geometry to algebra.\n- It enhances understanding of proportional relationships in shapes.\n- It’s essential for tackling more complex problems involving multiple variables.", "### Real-World Application", "Imagine designing a rectangular garden where the width is set to a specific measurement—say, ( w = 8 ) feet—and the length is three times that. Using the equation, you can quickly compute the exact perimeter: ( 2(8 + 24) = 64 ) feet, confirming your design matches the required space.", "### Conclusion", "Substituting length and width with symbolic expressions like ( 2(3w + w) = 64 ) transforms abstract geometry problems into clear, solvable equations. This approach strengthens algebraic reasoning and deepens comprehension of how variables represent real-world dimensions. Embrace this technique to simplify perimeter calculations and build a strong foundation for advanced mathematics.", "---", "Keywords: substitute expressions, length and width, perimeter equation, algebra geometry, solving perimeter problems, ( 2(3w + w) = 64 ), variable dimension, linear equations, real-world math applications."]