D'(x) = \frac{50}{16}x + 3 = \frac{25}{8}x + 3 = 0 \Rightarrow x = -\frac{24}{25}

["Solving the Linear Equation: x = –24/25 – A Step-by-Step Breakdown", "Understanding how to solve linear equations is a fundamental skill in algebra, essential for students and professionals alike. One such equation commonly encountered is:", "[\nD'(x) = \frac{50}{16}x + 3 = \frac{25}{8}x + 3 = 0\n]", "This expression leads directly to the solution ( x = -\frac{24}{25} ). In this article, we’ll explore how to solve this equation step-by-step, clarify common pitfalls, and explain why this value matters in mathematical applications.", "---", "### Step 1: Simplifying the Equation", "The given equation combines two equal expressions set equal to zero:", "[\n\frac{50}{16}x + 3 = \frac{25}{8}x + 3 = 0\n]", "Note: Both sides of an equation set to zero imply that each side equals zero. Therefore, we can work with either expression. For simplicity, we focus on one:", "[\n\frac{50}{16}x + 3 = 0\n]", "We will later confirm equivalence with the second form ( \frac{25}{8}x + 3 ), which has the same solution.", "---", "### Step 2: Eliminate Fractions for Easier Calculation", "Working with fractions can be cumbersome. Convert ( \frac{50}{16} ) to lowest terms:", "[\n\frac{50}{16} = \frac{25}{8}\n]", "So the equation becomes:", "[\n\frac{25}{8}x + 3 = 0\n]", "Subtract 3 from both sides:", "[\n\frac{25}{8}x = -3\n]", "---", "### Step 3: Solve for ( x )", "To isolate ( x ), multiply both sides by the reciprocal of ( \frac{25}{8} ), which is ( \frac{8}{25} ):", "[\nx = -3 \cdot \frac{8}{25} = -\frac{24}{25}\n]", "---", "### Step 4: Cross-Check with the Second Form", "To validate, solve:", "[\n\frac{25}{8}x + 3 = 0\n]", "Subtract 3:", "[\n\frac{25}{8}x = -3\n]", "Multiply both sides by ( \frac{8}{25} ):", "[\nx = -3 \cdot \frac{8}{25} = -\frac{24}{25}\n]", "This confirms the solution is consistent.", "---", "### Why ( x = -\frac{24}{25} ) Is Important", "This rational solution often appears in problems involving rates, linear modeling, and equilibrium conditions. For example, in physics, it might represent a balance point—such as when two forces or growth/decay trends intersect. Precise decimal or fractional forms ensure accuracy in calculations and interpretations.", "---", "### Final Summary", "The solution to the equation\n[\nD'(x) = \frac{50}{16}x + 3 = \frac{25}{8}x + 3 = 0\n]\nis:", "[\n\boxed{x = -\frac{24}{25}}\n]", "By simplifying fractions and carefully isolating the variable, we systematically arrive at this result—demonstrating clarity and precision in solving linear equations.", "---", "Key takeaways:\n- Always simplify fractions early to avoid errors.\n- Set each side of an equation equal to zero when solving linear expressions.\n- Multiplying by the reciprocal is crucial for isolating ( x ).\n- Verifying with alternative forms boosts confidence in the solution.", "If you’re learning algebra or teaching math, mastering this step-by-step process strengthens problem-solving skills essential across STEM disciplines.", "Keywords: solving linear equations, x = –24/25, algebra simplification, linear modeling, equation solver, fractional coefficients."]