Substitute \( l = 2w + 5 \) into the perimeter formula: \( 2(2w + 5) + 2w = 54 \).

["Substitute ( l = 2w + 5 ) into the Perimeter Formula: A Step-by-Step Solution", "When solving problems involving the perimeter of a rectangular shape, understanding how to substitute expressions correctly is essential for accurate results. One common scenario in algebra and geometry involves substituting a side length expressed in terms of another variable to find the perimeter. This article explains how to substitute ( l = 2w + 5 ) into the perimeter formula ( 2l + 2w = 54 ), simplifying the equation step by step.", "---", "### Understanding the Problem", "The perimeter ( P ) of a rectangle is given by:\n[\nP = 2l + 2w\n]\nwhere ( l ) is the length and ( w ) is the width. In this problem, we’re told that the length ( l ) is related to the width ( w ) by the equation:\n[\nl = 2w + 5\n]\nThe total perimeter is also given as 54 units:\n[\n2l + 2w = 54\n]", "Substituting ( l = 2w + 5 ) into the perimeter formula eliminates ( l ), allowing you to solve for ( w ), then find ( l ), and verify the solution.", "---", "### Step-by-Step Substitution", "Start with the perimeter equation:\n[\n2l + 2w = 54\n]", "Now substitute ( l = 2w + 5 ):\n[\n2(2w + 5) + 2w = 54\n]", "This is the core substitution step. Multiply 2 by each term inside the parentheses:\n[\n(2 \ imes 2w) + (2 \ imes 5) + 2w = 54\n\Rightarrow 4w + 10 + 2w = 54\n]", "Combine like terms (the ( w ) terms):\n[\n(4w + 2w) + 10 = 54\n\Rightarrow 6w + 10 = 54\n]", "Now solve for ( w ):\n[\n6w = 54 - 10\n\Rightarrow 6w = 44\n\Rightarrow w = \frac{44}{6} = \frac{22}{3}\n]", "---", "### Finding the Length", "Now substitute ( w = \frac{22}{3} ) back into ( l = 2w + 5 ):\n[\nl = 2\left(\frac{22}{3}\right) + 5 = \frac{44}{3} + 5 = \frac{44}{3} + \frac{15}{3} = \frac{59}{3}\n]", "---", "### Verification of the Solution", "Check the perimeter with these values:\n[\n2l + 2w = 2\left(\frac{59}{3}\right) + 2\left(\frac{22}{3}\right) = \frac{118}{3} + \frac{44}{3} = \frac{162}{3} = 54\n]\nThe perimeter calculation matches the given value of 54, confirming the solution is correct.", "---", "### Why This Substitution Matters", "Substituting algebraic expressions into formulas like the perimeter equation allows you to solve for unknown dimensions when one variable depends on another. This method is valuable in geometry, engineering, and real-world applications where relationships between dimensions are defined by equations.", "---", "### Conclusion", "By correctly substituting ( l = 2w + 5 ) into ( 2l + 2w = 54 ), simplifying step by step, we find the width ( w = \frac{22}{3} ) and length ( l = \frac{59}{3} ). This approach demonstrates a foundational algebraic technique that supports accurate problem-solving in mathematical modeling and geometry.", "---", "Keywords: substitute ( l = 2w + 5 ) into perimeter formula, algebra substitution, perimeter calculation, solve for width and length, geometry problem solving, linear equations, rectangular perimeter.", "---", "Note: This method is efficient for equations involving one unknown expressed in terms of another—ideal for students, teachers, and practitioners working with algebraic geometry problems."]