["Solving the Equation: ( 2(2w + 3) + 2w = 54 )", "Solving algebraic equations is a fundamental skill in mathematics, and understanding how to simplify and solve linear equations opens the door to more complex problem-solving. One common problem many learners encounter is solving equations like ( 2(2w + 3) + 2w = 54 ). Whether you're a student, a parent, or a math enthusiast, mastering this type of equation enhances your algebraic proficiency.", "In this comprehensive guide, we’ll walk step-by-step through solving the equation ( 2(2w + 3) + 2w = 54 ), explain key algebraic concepts, and provide tips for efficient problem-solving.", "---", "### Step 1: Expand the Parentheses
\nThe first step is to eliminate the parentheses by distributing the 2 across ( 2w + 3 ):", "[
\n2(2w + 3) = 2 \cdot 2w + 2 \cdot 3 = 4w + 6
\n]", "Now substitute back into the original equation:", "[
\n4w + 6 + 2w = 54
\n]", "---", "### Step 2: Combine Like Terms
\nCombine the ( w )-terms and constant terms:", "[
\n(4w + 2w) + 6 = 54 \implies 6w + 6 = 54
\n]", "This simplifies solving significantly by reducing the number of variables.", "---", "### Step 3: Isolate the Variable
\nSubtract 6 from both sides to isolate the term with ( w ):", "[
\n6w + 6 - 6 = 54 - 6 \implies 6w = 48
\n]", "---", "### Step 4: Solve for ( w )
\nDivide both sides by 6:", "[
\nw = \frac{48}{6} = 8
\n]", "---", "### Final Answer
\nThe solution to the equation ( 2(2w + 3) + 2w = 54 ) is:", "[
\n\boxed{w = 8}
\n]", "---", "### Why This Equation Matters
\nEquation solving like ( 2(2w + 3) + 2w = 54 ) builds foundational algebraic fluency. Real-world applications include budgeting, physics (like motion calculations), and computer programming, where variables represent dynamic quantities.", "---", "### Tips to Solve Similar Equations Easily
\n- Expand parentheses carefully: Always apply the distributive property to avoid sign errors.
\n- Group like terms: Combine ( w )-terms and constant terms to simplify.
\n- Isolate the variable: Use addition/subtraction before multiplication/division.
\n- Check your solution: Substitute ( w = 8 ) into the original equation to verify correctness:", "[
\n2(2 \cdot 8 + 3) + 2 \cdot 8 = 2(16 + 3) + 16 = 2 \cdot 19 + 16 = 38 + 16 = 54
\n]", "✅ Identity confirmed—your solution is correct!", "---", "Mastering equations like ( 2(2w + 3) + 2w = 54 ) strengthens algebraic reasoning and prepares learners for advanced math topics. Keep practicing, stay patient, and watch your problem-solving skills grow!"]
