\( 150 = 5(10 + 9d) \)

["Solving the Equation ( 150 = 5(10 + 9d) ): A Step-by-Step Guide", "Understanding how to solve equations like ( 150 = 5(10 + 9d) ) is essential for students, educators, and anyone working with algebra. This equation appears frequently in math classrooms and is fundamental for mastering linear relationships. In this article, we’ll break down how to solve for ( d ), explain the algebra behind it, and highlight its real-world relevance.", "---", "### Understanding the Equation", "The equation\n[ 150 = 5(10 + 9d) ]\nis a linear equation where one side equals a constant (150), and the other side is 5 multiplied by a binomial ( (10 + 9d) ). Solving it means finding the value of ( d ) that makes both sides equal.", "---", "### Step-by-Step Solution", "Step 1: Simplify the right-hand side\nDistribute the 5 across the terms inside the parentheses:\n[ 150 = 5 \cdot 10 + 5 \cdot 9d ]\n[ 150 = 50 + 45d ]", "Step 2: Isolate the variable term\nSubtract 50 from both sides to eliminate the constant on the right:\n[ 150 - 50 = 45d ]\n[ 100 = 45d ]", "Step 3: Solve for ( d )\nDivide both sides by 45:\n[ d = \frac{100}{45} ]\nSimplify the fraction by dividing numerator and denominator by 5:\n[ d = \frac{20}{9} ]", "---", "### Final Answer", "The solution to the equation $ 150 = 5(10 + 9d) $ is\n[ d = \frac{20}{9} ]", "Approximately, ( d \approx 2.22 ), meaning each unit ( d ) represents roughly 2.22 contributes to reaching the total of 150 in this proportional relationship.", "---", "### Real-World Applications", "Equations like this model real-life scenarios:\n- Budgeting, where ( d ) might represent units of an item purchased over a fixed cost (e.g., $50 fixed fee plus $9 per item).\n- Physics problems involving motion or force with linear scaling.\n- Economics and supply-demand models where variables represent pricing and quantity.", "---", "### Why Learning This Equation Matters", "Mastering solving linear equations strengthens critical thinking and problem-solving skills. Understanding how constants, coefficients, and variables interact equips learners for advanced math, STEM careers, and practical decision-making.", "---", "### Tips for Practicing Linear Equations", "- Always simplify both sides of the equation before solving.\n- Isolate the variable term step-by-step.\n- Check your solution by plugging ( d = \frac{20}{9} ) back into the original equation to confirm equality.\n- Use graphs or tables to visualize the equation’s behavior.", "---", "Conclusion", "The equation ( 150 = 5(10 + 9d) ) is a clear example of linear algebra with wide-ranging applications. By following systematic steps—distributing, isolating, and simplifying—you can confidently solve for any unknown. Whether in homework, standardized tests, or professional fields, equations help translate real problems into algebraic solutions. Keep practicing, stay curious, and let algebra unlock new insights!", "---", "Keywords: equation solving, linear equations, algebra tutorial, solve for d, math practice, linear expression, ( 150 = 5(10 + 9d) ) solution, equation steps, math fundamentals, fractional solutions, proportional reasoning."]