\( 2((2w + 5) + w) = 60 \).

["## Solving the Equation: ( 2((2w + 5) + w) = 60 )", "Understanding how to solve linear equations is essential for mastering algebra and building strong problem-solving skills. The equation ( 2((2w + 5) + w) = 60 ) is a great example of a seemingly complex expression that simplifies neatly into a standard quadratic form. In this article, we’ll break down step-by-step how to solve ( 2((2w + 5) + w) = 60 ), explain the logic behind each step, and highlight why this type of problem matters in mathematics and real-world applications.", "---", "### What Is the Equation ( 2((2w + 5) + w) = 60 ) All About?", "At first glance, the equation contains nested parentheses and coefficients, but it simplifies into a solvable linear expression. It represents a situation where a value (scaled by 2) is applied to the sum of ( 2w + 5 ) and ( w ). Solving this equation reveals the value of the variable ( w ), critical for both academic learning and practical applications like resource planning, budgeting, and engineering calculations.", "---", "### Step-by-Step Solution", "#### Step 1: Simplify the expression inside the parentheses\nStart with the inner parentheses:\n[\n(2w + 5) + w\n]\nCombine like terms:\n[\n2w + w + 5 = 3w + 5\n]\nSo the equation becomes:\n[\n2(3w + 5) = 60\n]", "#### Step 2: Distribute the 2\nMultiply both sides of the equation by 2 to eliminate the coefficient:\n[\n2(3w + 5) \ imes 2 = 60 \ imes 2 \quad \Rightarrow \quad 4(3w + 5) = 120\n]", "#### Step 3: Distribute again\nApply the distributive property:\n[\n4 \cdot 3w + 4 \cdot 5 = 120 \quad \Rightarrow \quad 12w + 20 = 120\n]", "#### Step 4: Isolate the variable term\nSubtract 20 from both sides:\n[\n12w + 20 - 20 = 120 - 20 \quad \Rightarrow \quad 12w = 100\n]", "#### Step 5: Solve for ( w )\nDivide both sides by 12:\n[\nw = \frac{100}{12} = \frac{25}{3}\n]\nSimplify the fraction:\n[\nw = 8\frac{1}{3} \quad \ ext{(optional but insightful)}\n]", "---", "### Final Answer", "[\n\boxed{w = \frac{25}{3}}\n]", "---", "### Why This Equation Matters", "Solving equations like ( 2((2w + 5) + w) = 60 ) builds foundational skills in algebra, including:\n- Order-of-operations application\n- Distributive property use\n- Fraction manipulation\n- Stepwise problem decomposition", "Real-world uses include determining break-even points in economics, calculating dimensions in geometry, or balancing formulae in science. Understanding how to manipulate nested expressions also prepares learners for more advanced topics like polynomial equations and systems of equations.", "---", "### Practice Problem: Try It Yourself!", "Solve the equation:\n[\n3(2z - 4 + z) = 45\n]", "Follow the same steps: simplify parentheses, distribute, isolate the variable—and verify your solution.", "---", "### Conclusion", "The equation ( 2((2w + 5) + w) = 60 ) is a manageable yet powerful example of solving linear expressions through systematic simplification. By mastering these techniques, you strengthen your algebraic reasoning—and unlock the ability to tackle complex problems across science, engineering, finance, and beyond. Keep practicing, stay curious, and watch your confidence in math grow!"]