Vereinfache: \( C = 50x + \frac{2550 - 300x}{7} \).
["Optimize Your Calculations: Simplify the Equation ( C = 50x + \frac{2550 - 300x}{7} )", "When working with linear equations in algebra or real-world applications—such as cost modeling, budgeting, or production planning—it’s often necessary to simplify expressions for clarity and easier analysis. One common form you may encounter is:", "[\nC = 50x + \frac{2550 - 300x}{7}\n]", "This equation represents a linear relationship where ( C ) stands for a total cost, and ( x ) typically represents a variable quantity—like hours worked, units produced, or another controllable factor.", "### Why Simplify This Equation?", "Simplifying allows you to:", "- Understand the relationship between variables at a glance\n- Identify key coefficients affecting total cost\n- Easily analyze by substituting ( x ) values\n- Prepare for graphing or further algebraic manipulation", "---", "### Step-by-Step Simplification", "Start with:", "[\nC = 50x + \frac{2550 - 300x}{7}\n]", "We separate the fraction to make terms clearer:", "[\nC = 50x + \frac{2550}{7} - \frac{300x}{7}\n]", "Now combine like terms—specifically the terms involving ( x ):", "[\nC = \left(50x - \frac{300x}{7}\right) + \frac{2550}{7}\n]", "Express ( 50x ) as a fraction with denominator 7:", "[\n50x = \frac{350x}{7}\n]", "So:", "[\nC = \frac{350x}{7} - \frac{300x}{7} + \frac{2550}{7}\n]", "[\nC = \frac{(350x - 300x)}{7} + \frac{2550}{7}\n]", "[\nC = \frac{50x}{7} + \frac{2550}{7}\n]", "Factor out ( \frac{50}{7} ) from the first term (optional):", "[\nC = \frac{50x + 2550}{7}\n]", "---", "### Final Simplified Form", "[\n\boxed{C = \frac{50x + 2550}{7}}\n]", "This compact form reveals directly how total cost ( C ) depends on ( x ) through a linear numerator scaled by ( \frac{1}{7} ), making forecasting and optimization straightforward.", "---", "### Practical Applications", "- Cost Analysis: Ideal for businesses calculating variable costs based on units or labor hours.\n- Budgeting Models: Helps in quickly estimating total expenses as operational variables change.\n- Educational Tools: Simplifies teaching algebra by showing how rational expressions reduce into simpler linear forms.", "---", "### Conclusion", "Simplifying the expression ( C = 50x + \frac{2550 - 300x}{7} ) to ( \boxed{C = \frac{50x + 2550}{7}} ) transforms a complex fractional expression into a clear, usable linear equation. This simplification enhances clarity, supports efficient computation, and aids in modeling real-life financial relationships with precision.", "---", "Keywords: Simplify linear equation, algebraic simplification, cost function algebra, rational expression simplification, algebra teaching resource, financial modeling equation, C = (50x + 2550)/7, equation simplification guide\nMeta Description: Simplify ( C = 50x + \frac{2550 - 300x}{7} ) to ( C = \frac{50x + 2550}{7} ) for clearer cost modeling. Easy step-by-step explanation with practical applications."]
