Their sum: \( x + (x+2) + (x+4) = 78 \).

["Understanding the Equation: Solving ( x + (x+2) + (x+4) = 78 )", "When combined, algebraic expressions like ( x + (x+2) + (x+4) = 78 ) provide an efficient way to solve real-world problems involving unknowns. In this article, we’ll break down how to solve the equation step-by-step, explain its significance, and explore its applications. Whether you’re a high school student, educator, or math enthusiast, mastering this sum is essential for building algebraic skills.", "---", "### What is the Equation?", "Consider the equation:\n[ x + (x + 2) + (x + 4) = 78 ]", "On the left side, we sum three linear expressions: a variable ( x ), another variable incremented by 2, and a third incremented by 4. The right-hand side is a constant equal to 78. Solving this equation finds the specific value of ( x ) that balances the equation.", "---", "### Step-by-Step Solution", "Let’s simplify and solve step-by-step:", "1. Expand the expression:\n [\n x + (x + 2) + (x + 4) = x + x + 2 + x + 4\n ]", "2. Combine like terms:\n Add up the coefficients of ( x ) and constants:\n [\n (x + x + x) + (2 + 4) = 3x + 6\n ]", "3. Set equation equal to 78:\n [\n 3x + 6 = 78\n ]", "4. Isolate the variable term:\n Subtract 6 from both sides:\n [\n 3x = 72\n ]", "5. Solve for ( x ):\n Divide both sides by 3:\n [\n x = 24\n ]", "---", "### Verifying the Solution", "Plug ( x = 24 ) back into the original equation to confirm:\n[\n24 + (24 + 2) + (24 + 4) = 24 + 26 + 28 = 78\n]", "Since the left side equals 78, the solution is correct.", "---", "### Why This Equation Matters", "Solving equations like ( x + (x+2) + (x+4) = 78 ) builds foundational algebra skills used in:", "- Word problems: Interpreting and modeling real-life situations involving totals.\n- Data analysis: Finding unknowns in linear datasets.\n- STEM applications: Designing equations in physics, economics, and engineering.\n- Problem-solving strategy: Training logical reasoning and step-by-step analysis.", "---", "### Alternative View: Simplifying the Sum First", "Notice the terms form an arithmetic sequence:\n( x ), ( x+2 ), ( x+4 ) — common difference of 2. The sum of three equally spaced terms is three times the middle term. Here, the middle term is ( x+2 ). Thus:\n[\nx + (x+2) + (x+4) = 3(x + 2)\n]\nSet equal to 78:\n[\n3(x + 2) = 78\n]\nDivide both sides by 3:\n[\nx + 2 = 26 \Rightarrow x = 24\n]", "This shortcut avoids expanding and combines insight with algebra.", "---", "### Practical Applications and Extensions", "For learners looking beyond the solution:", "- Pattern recognition: The sum ( x + (x+2) + (x+4) ) models cumulative totals increasing uniformly.\n- Generalization: Similarly, any three consecutive terms with constant difference can use the average-based shortcut.\n- Graphical interpretation: The equation represents a linear function intersecting the value 78.", "---", "### Conclusion", "Solving ( x + (x+2) + (x+4) = 78 ) isn’t just about finding ( x = 24 ); it’s about learning to simplify complexity, verify logic, and apply algebraic reasoning. Whether you’re tackling this in class or solving real-world problems, mastering such equations strengthens your mathematical toolkit and confidence.", "Remember:\n[\n\boxed{x = 24}\n]\nis the verified solution to the equation.", "---", "Related Keywords:\n( x + (x+2) + (x+4) = 78 solution, how to solve linear equations, algebraic word problems, solving arithmetic sequences algebraically, step-by-step equation solving, primary algebra education, Russian math olympiad techniques (adaptable), solving systems with scaling methods.", "---", "Start building your algebra mastery today — with a simple but powerful equation as your guide!"]