\[ 150 = \frac{10}{2} (2 \times 5 + 9d) \]

["Solving the Equation 150 = (10 / 2)(2 × 5 + 9d): A Complete Guide", "Understanding how to solve algebraic equations is essential for mastering math concepts and building strong problem-solving skills. One challenging equation that often appears in algebra studies is:", "[ 150 = \frac{10}{2} (2 \ imes 5 + 9d) ]", "In this article, we’ll break down how to solve this equation step-by-step, explain the key algebraic principles involved, and show how this expression connects to real-world applications and deeper mathematical concepts.", "---", "### Step-by-Step Solution to the Equation", "Let’s solve:\n[ 150 = \frac{10}{2} (2 \ imes 5 + 9d) ]", "1. Simplify constants on the right side\n First, compute (\frac{10}{2}):\n [\n \frac{10}{2} = 5\n ]\n Substitute back:\n [\n 150 = 5 (2 \ imes 5 + 9d)\n ]", "2. Multiply inside the parentheses\n Calculate (2 \ imes 5):\n [\n 2 \ imes 5 = 10\n ]\n So the equation becomes:\n [\n 150 = 5 (10 + 9d)\n ]", "3. Distribute the 5\n Multiply 5 into the parentheses:\n [\n 5 \ imes 10 = 50,\quad 5 \ imes 9d = 45d\n ]\n Now:\n [\n 150 = 50 + 45d\n ]", "4. Isolate the variable term\n Subtract 50 from both sides:\n [\n 150 - 50 = 45d \quad \Rightarrow \quad 100 = 45d\n ]", "5. Solve for (d)\n Divide both sides by 45:\n [\n d = \frac{100}{45} = \frac{20}{9}\n ]", "---", "### Final Answer\n[\n\boxed{d = \frac{20}{9}}\n]", "---", "### Why This Equation Matters: Algebraic Foundations", "At first glance, the equation ( 150 = \frac{10}{2}(2 \ imes 5 + 9d) ) may seem like a random math problem, but it reveals fundamental algebraic processes:", "- Order of operations (PEMDAS/BODMAS): Properly simplifying expressions using parentheses, multiplication, and division ensures accuracy.\n- Distributive property: Recognizing how multiplication distributes over addition within parentheses.\n- Variable isolation: The ability to isolate unknowns is a critical skill in equation solving, essential in physics, engineering, and finance.", "---", "### Real-World Applications", "Equations like this often appear in modeling scenarios such as:\n- Budgeting: Calculating how much of a variable cost (e.g., advertising expenses) fits within a fixed budget.\n- Physics: Modeling relationships between distance, speed, and time, where variables represent unknowns needing determination.\n- Data analysis: Determining a missing multiplier in proportional relationships, crucial in economics and research.", "---", "### Expanded Mathematical Insight", "Let’s interpret the structure:", "- The left side, a constant 150, represents a fixed total.\n- The right side combines constants and a variable (d) in a multiplicative setup.\n- The equation models a balance: what value of (d) will balance the left side, given distributed constants and coefficients?", "This pattern is common in real-world problems where fixed and variable components must match exactly, making such equations vital tools in problem-solving.", "---", "### Practice Tips", "To master solving similar equations:\n1. Always follow order of operations carefully.\n2. Simplify both sides before isolating variables.\n3. Double-check substitutions and arithmetic at each step.\n4. Translate word problems into symbolic form to build intuition.", "---", "Conclusion", "Solving ( 150 = \frac{10}{2}(2 \ imes 5 + 9d) ) is more than just algebra — it’s a gateway to logical reasoning and quantitative problem-solving. By breaking down each step, understanding the principles at play, and recognizing its real-world relevance, students and learners of all levels can strengthen their mathematical foundation.", "Key takeaway:\nNever underestimate the power of carefully analyzing an equation — each operation is a clue to unlocking the solution.", "---", "Related Search Terms:\n- How to solve linear equations step by step\n- Algebraic equations with variables on both sides\n- Understanding proportional relationships in math\n- Step-by-step solving of equations with fractions", "---", "Equation Tip of the Day:\nAlways simplify constants and apply distribution correctly — they often make the difference between a confusing mess and a clear path to the answer.", "---", "Understanding this equation today empowers you to tackle more complex math tomorrow — from calculus to economics. Keep practicing, questioning, and solving!"]