\(0.20x + 15 - 0.50x = 10.5 \Rightarrow -0.30x = -4.5 \Rightarrow x = 15\)

["Solving the Linear Equation: Step-by-Step Guide to (0.20x + 15 - 0.50x = 10.5)", "Mastering linear equations is essential in algebra, and solving equations like (0.20x + 15 - 0.50x = 10.5) may seem complex at first, but it follows a clear, logical sequence. This article walks you through solving this equation step-by-step, why it works, and why understanding these fundamentals matters in everyday math, science, and algebra-heavy schoolwork.", "---", "### What Is the Equation?", "We start with:\n[\n0.20x + 15 - 0.50x = 10.5\n]", "This linear equation combines constant terms (15), variable terms ((0.20x, -0.50x)), and equals a number (10.5). Solving such equations helps find unknowns like (x), which is crucial in physics, economics, engineering, and data analysis.", "---", "### Step 1: Combine Like Terms", "The first step is combining the like terms on the left-hand side. Here, (0.20x) and (-0.50x) both contain (x):", "[\n(0.20x - 0.50x) + 15 = 10.5\n]", "Calculate:\n[\n-0.30x + 15 = 10.5\n]", "This simplification reduces complexity while preserving the equation’s integrity.", "---", "### Step 2: Isolate the Variable Term", "Next, subtract 15 from both sides to move the constant to the right:", "[\n-0.30x = 10.5 - 15\n]", "[\n-0.30x = -4.5\n]", "This isolates (x) multiplied by the coefficient (-0.30), making it easier to solve for (x).", "---", "### Step 3: Solve for (x)", "Divide both sides by (-0.30):", "[\nx = \frac{-4.5}{-0.30}\n]", "Simplify the division:\n[\nx = 15\n]", "The equation confirms that when (x = 15), the original expression balances.", "---", "### Understanding the Solution: Why (x = 15) Works", "Plug (x = 15) back into the original equation:", "[\n0.20(15) + 15 - 0.50(15) = 3 + 15 - 7.5 = 10.5\n]", "The left side equals the right side, confirming our solution is correct.", "---", "### Real-World Implications", "Equations like this appear in real-life applications:\n- Finance: Calculating break-even points\n- Physics: Determining when two linear motions intersect\n- Engineering: Modeling material stress under load", "Learning how to solve them builds problem-solving confidence and precision—skills vital in STEM fields and beyond.", "---", "### Final Summary", "To summarize solving (0.20x + 15 - 0.50x = 10.5):", "1. Combine like terms: (-0.30x + 15 = 10.5)\n2. Isolate variable: (-0.30x = -4.5)\n3. Solve: (x = 15)", "This clear, stepwise approach not only finds the answer but also strengthens algebraic reasoning—key for academic success and real-world problem solving.", "---", "Key Takeaway: Like solving any linear equation, consistent practice and systematic steps simplify complex problems. Whether you’re a student learning algebra or a professional applying equations, mastering this technique is invaluable. Start with like terms, isolate variables, and verify your solution—your math confidence will grow with each step."]