\( 120x + 180y = 7800 \)
["Mastering the Equation: Understanding and Solving (120x + 180y = 7800) for Optimal Solutions", "The linear Diophantine equation (120x + 180y = 7800) appears deceptively simple at first glance, but behind this standard form lies a powerful tool used in various real-world applications—from budget planning and resource allocation to optimization problems in engineering and economics. In this SEO-optimized article, we’ll explore how to solve, interpret, and apply this equation using methods that are both mathematically sound and practically relevant.", "---", "### Introduction to the Equation\nThe equation (120x + 180y = 7800) is a linear diophantine equation involving two variables, (x) and (y), representing quantities of two variables constrained by proportional coefficients. Our goal is to find all integer solutions ((x, y)) that satisfy the equation, interpreting them within the context of real-life scenarios.", "---", "### Step 1: Simplify the Equation for Easier Computation\nBefore solving, simplify the equation by dividing all terms by their greatest common divisor (GCD). The coefficients 120, 180, and 7800 share a common factor:", "[\n\ ext{GCD}(120, 180) = 60\n]", "Divide each term by 60:", "[\n\frac{120x}{60} + \frac{180y}{60} = \frac{7800}{60}\n\Rightarrow 2x + 3y = 130\n]", "Now we work with the simplified form:", "[\n2x + 3y = 130\n]", "This simplified equation is easier to analyze and solve for integer values.", "---", "### Step 2: Solve the Diophantine Equation\nWith (2x + 3y = 130), this is a linear Diophantine equation. To find integer solutions:", "- Since GCD(2, 3) = 1, and 1 divides 130, integer solutions exist.", "#### General Solution Form\nWe solve for one variable and express the general solution.", "Let’s solve for (x):", "[\n2x = 130 - 3y \Rightarrow x = \frac{130 - 3y}{2}\n]", "For (x) to be an integer, (130 - 3y) must be even. Since 130 is even, (3y) must be even. As 3 is odd, (y) must be even.", "Let (y = 2k) for integer (k). Substitute:", "[\nx = \frac{130 - 3(2k)}{2} = \frac{130 - 6k}{2} = 65 - 3k\n]", "Thus, the complete general integer solution is:", "[\nx = 65 - 3k,\quad y = 2k,\quad \ ext{where } k \in \mathbb{Z}\n]", "---", "### Step 3: Find Non-Negative Integer Solutions\nIn practical applications—such as budgeting or inventory—negatives or non-physical values are invalid. So we restrict (x \geq 0) and (y \geq 0):", "[\nx = 65 - 3k \geq 0 \Rightarrow k \leq \frac{65}{3} \approx 21.67 \Rightarrow k \leq 21\n]", "[\ny = 2k \geq 0 \Rightarrow k \geq 0\n]", "Combining: (0 \leq k \leq 21)", "Thus, valid integer values for (k) range from 0 to 21, inclusive—giving 22 valid solution pairs.", "---", "### Step 4: Practical Interpretations and Applications\nThe equation models scenarios like:", "- Budget Allocation: Suppose (x) and (y) represent quantities of two products costing $120 and $180 respectively, and the total budget is $7800. The equation defines all combinations of products that match the budget.", "- Resource Distribution: In manufacturing, (x) and (y) may represent units of two materials required for an assembly, limited by a total material cost.", "- Time Management Problems: When (x) and (y) are effort units of two tasks with different time costs and a fixed total time or resource.", "Each solution ((x, y)) corresponds to a feasible strategy within constraints.", "---", "### Step 5: How to Use This in Real Life\nTo apply (120x + 180y = 7800) (or its simplified form (2x + 3y = 130)):", "1. Define real-world units clearly for (x) and (y).\n2. Use the general solution (x = 65 - 3k), (y = 2k) with (k = 0) to (21).\n3. Filter non-negative values to ensure practical feasibility.\n4. Analyze trade-offs—small changes in (k) reflect alternative resource allocations.", "---", "### Conclusion: The Power of Linear Equations in Problem Solving\nThe equation (120x + 180y = 7800) exemplifies how a simple linear model enables structured, integer-based decision-making. By simplifying and systematically solving for variable pairs, you unlock clarity and precision in diverse applications—from business planning to educational projects.", "Mastering such equations empowers strong analytical reasoning, essential for students, engineers, economists, and data analysts alike.", "---", "### SEO Keywords for the Article\n- Solving (120x + 180y = 7800)\n- Diophantine equation solutions\n- Integer solutions for linear equations\n- How to solve (2x + 3y = 130)\n- Applications of (120x + 180y = 7800 in real life\n- General solution (x = 65 - 3k), (y = 2k)\n- Budgeting with linear equations\n- Optimization using integer programming", "---", "Keywords summary: This SEO-optimized article covers solving the key Diophantine equation, simplification steps, general solution derivation, value constraints, and real-world applications—making it highly relevant for learners and professionals seeking clarity in applied mathematics.", "---", "Stay insightful. Solve intelligently.\nFor more articles on linear equations, integer solutions, and practical math applications, explore our full library on mathematical problem solving."]
