Perimeter equation: \( 2(w + (2w + 4)) = 68 \)

["Understanding the Perimeter Equation: Solving ( 2(w + (2w + 4)) = 68 )", "In algebra, solving equations involving expressions for geometric shapes helps build strong problem-solving skills. One common example is finding the perimeter of a rectangle using a variable for one side length. An important step in such problems is correctly simplifying perimeter equations — a skill mastered through equations like ( 2(w + (2w + 4)) = 68 ).", "### What is the Perimeter Equation?", "The perimeter of a rectangle is calculated using the formula:\n[\n\ ext{Perimeter} = 2 \ imes (\ ext{length} + \ ext{width})\n]\nIn this equation, suppose the width is ( w ) and the length is ( 2w + 4 ). Substituting these into the perimeter formula gives:\n[\n2(w + (2w + 4)) = 68\n]\nThis equation models the relationship between the width and the total perimeter.", "### Step-by-Step Solution", "1. Simplify inside the parentheses:\nStart by simplifying the expression inside the parentheses:\n[\nw + (2w + 4) = w + 2w + 4 = 3w + 4\n]\nSo the equation becomes:\n[\n2(3w + 4) = 68\n]", "2. Distribute the 2:\nMultiply through by 2:\n[\n6w + 8 = 68\n]", "3. Isolate the variable ( w ):\nSubtract 8 from both sides:\n[\n6w = 60\n]\nThen divide by 6:\n[\nw = 10\n]", "4. Find the length:\nSubstitute ( w = 10 ) back into the expression for the length:\n[\n2w + 4 = 2(10) + 4 = 20 + 4 = 24\n]", "5. Calculate the perimeter (optional check):\nVerify by plugging values back:\n[\n2(w + \ ext{length}) = 2(10 + 24) = 2 \ imes 34 = 68\n]\nThis confirms the solution is correct.", "### Why Solve This Type of Equation?", "Such perimeter equations help students develop algebra fluency by combining:\n- Parentheses\n- Like terms\n- Multiplication and division\n- Inverse operations to isolate variables", "They also connect algebra to real-world applications, such as determining fencing required for a plot or material costs for rectangular structures.", "### Bonus: Practice Tips", "- Always simplify inside parentheses first.\n- Distribute carefully and watch for common mistakes like sign errors.\n- Test your solution by substituting ( w = 10 ) into the original equation.\n- This type of problem strengthens logical reasoning and algebra fundamentals.", "---", "Keywords: perimeter equation, solve ( 2(w + (2w + 4)) = 68 ), algebraic equation solving, rectangle perimeter algebra, step-by-step math, algebraic expressions, solving linear equations, math problem solving, solving equations with parentheses.", "By mastering equations like ( 2(w + (2w + 4)) = 68 ), learners lay a solid foundation for more complex geometry and algebraic applications."]