$$ 4(15) + 2(30) + c = 120 \Rightarrow 60 + 60 + c = 120 \Rightarrow c = 0 $$
["Understanding the Equation: $$(4 \ imes 15) + (2 \ imes 30) + c = 120$$ – A Simple Step-by-Step Solution", "Mathematics often appears intimidating at first glance, but many equations simplify cleanly with careful step-by-step reasoning. One such example is the equation:", "$$\n(4 \ imes 15) + (2 \ imes 30) + c = 120\n$$", "This seemingly simple expression involves basic arithmetic, substitution, and logic—all essential skills in problem-solving across everyday life and academic settings. Let’s break down how to solve it clearly and why the final result is $ c = 0 $.", "---", "### Step 1: Calculate the Multiplications", "Start by evaluating the products inside the equation.", "- $ 4 \ imes 15 = 60 $\n- $ 2 \ imes 30 = 60 $", "Substitute these values back into the equation:", "$$\n60 + 60 + c = 120\n$$", "---", "### Step 2: Combine Like Terms", "Next, add the two constants:", "$$\n60 + 60 = 120\n$$", "Now the equation becomes:", "$$\n120 + c = 120\n$$", "---", "### Step 3: Isolate the Variable", "To find $ c $, subtract 120 from both sides:", "$$\nc = 120 - 120\n$$", "$$\nc = 0\n$$", "---", "### What Does $ c = 0 $ Mean?", "In mathematics, assigning $ c = 0 $ means that no additional value is needed beyond the calculated total. In real-world terms, this equation models a balance—adding known quantities perfectly equals a target total. Here, the initial terms sum exactly to the whole, leaving zero residual.", "---", "### Why This Equation Matters", "This type of linear equation is foundational in algebra and essential for modeling linear relationships. From budgeting and business planning to science and engineering, understanding how to isolate variables and verify solutions builds logical thinking and quantitative reasoning.", "Embracing each step—calculating, simplifying, and isolating—helps demystify seemingly complex problems. Solving $(4 \ imes 15) + (2 \ imes 30) + c = 120$ step-by-step confirms conclusively that $ c = 0 $.", "---", "Key Takeaways:", "- Always simplify calculations first (multiplications, additions).\n- Substitute and combine terms carefully.\n- Use inverse operations to isolate unknowns.\n- Solutions validate logical consistency in equations.", "By mastering such step-by-step methods, you strengthen your mathematical foundation for future learning and real-life applications.", "---", "Need more help with algebra? Explore our guides on solving linear equations, variables, and step-by-step problem-solving techniques!"]
