\((2x + 3y) + (12x - 3y) = 12 + 30\) → \(14x = 42\) → \(x = 3\).

["# Solving ((2x + 3y) + (12x - 3y) = 12 + 30) †(14x = 42) †(x = 3): A Step-by-Step Breakdown", "Solving linear equations is a foundational skill in algebra, crucial for students, scientists, engineers, and anyone working with mathematical modeling. One common type of problem involves combining like terms and isolating variables. In this article, we break down the equation:", "[\n(2x + 3y) + (12x - 3y) = 12 + 30 \quad \ ext{â€} \quad 14x = 42 \quad \ ext{âaceq} \quad x = 3\n]", "We’ll explore how to simplify, solve, and understand each step to build confidence and clarity in algebraic reasoning.", "---", "## Step 1: Simplify Both Sides of the Equation", "Start with the original equation:", "[\n(2x + 3y) + (12x - 3y) = 12 + 30\n]", "First, simplify the right-hand side:", "[\n12 + 30 = 42\n]", "Now rewrite the equation:", "[\n(2x + 3y) + (12x - 3y) = 42\n]", "---", "## Step 2: Combine Like Terms on the Left Side", "Look for terms with the same variable or constant:", "- (2x + 12x = 14x)\n- (3y - 3y = 0)", "So the left-hand side simplifies to:", "[\n14x\n]", "Now your equation becomes:", "[\n14x = 42\n]", "---", "## Step 3: Solve for (x)", "To isolate (x), divide both sides by 14:", "[\nx = \frac{42}{14} = 3\n]", "---", "## Why Do the (y) Terms Cancel Out?", "Notice in the original expression:", "[\n2x + 3y + 12x - 3y\n]", "The (+3y) and (-3y) terms cancel each other:", "[\n3y - 3y = 0\n]", "This shows a key algebraic principle: terms with opposite coefficients sum to zero. This simplification was essential to reduce the equation to a single variable.", "---", "## Verifying the Solution", "Plug (x = 3) back into the original equation to confirm:", "Left side:", "[\n(2(3) + 3y) + (12(3) - 3y) = (6 + 3y) + (36 - 3y) = 42\n]", "Right side:", "[\n12 + 30 = 42\n]", "Both sides equal 42, confirming the solution is correct.", "---", "## Real-World Applications", "Equations like this appear in physics, economics, and engineering — for example:", "- Balancing forces or torques in mechanics\n- Equating total revenue and costs in business models\n- Solving for unknown parameters under system constraints", "Understanding how terms combine and simplify empowers problem-solving across disciplines.", "---", "## Summary: Key Takeaways", "- Combine like terms first to simplify expressions.\n- Opposite terms (e.g., (3y) and (-3y)) cancel during simplification.\n- Keep track of constants carefully (e.g., (12 + 30 = 42)).\n- Isolating the variable yields a clear, concise solution.", "By mastering these steps, you’ll confidently tackle equations involving variables, simplify complex expressions, and apply algebra effectively in real-world scenarios—starting from ((2x + 3y) + (12x - 3y) = 12 + 30) †(14x = 42) âç (x = 3).", "---", "Related Keywords for SEO:\nalgebra simplification, solving linear equations, canceling like terms, isolate variable, step-by-step equation solving, (14x = 42), linear equation solution, algebra tutorial, mathematical reasoning, solving equations with variables."]