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

["### Solve ( 150 = \frac{10}{2}(2 \ imes 5 + 9d) ): A Step-by-Step Algebra Guide", "If you've stumbled upon the equation\n[\n150 = \frac{10}{2}(2 \ imes 5 + 9d),\n]\nyou're not alone—many students and self-learners face similar algebraic challenges. This article breaks down how to solve this equation step-by-step, explaining key algebraic principles and algebraic manipulations needed to isolate the variable ( d ), ultimately revealing its value. Understanding this equation isn’t just about finding a number; it’s about mastering foundational algebra skills essential for solving more complex expressions.", "---", "### Understanding the Equation: Structure and Simplification", "Start by decoding the equation:\n[\n150 = \frac{10}{2}(2 \ imes 5 + 9d)\n]\nThe fraction (\frac{10}{2}) simplifies easily to (5), turning the equation into:\n[\n150 = 5(2 \ imes 5 + 9d)\n]", "Now think of this structure: a constant multiplied by a parenthetical expression. This form is common when solving for unknowns inside parentheses—often called factoring or distribution in algebra.", "---", "### Step 1: Simplify the parentheses expression", "First, calculate (2 \ imes 5 = 10). The equation becomes:\n[\n150 = 5(10 + 9d)\n]", "Distributing the 5 across the parentheses, we have:\n[\n150 = 5 \ imes 10 + 5 \ imes 9d = 50 + 45d\n]", "So the equation is now:\n[\n150 = 50 + 45d\n]", "---", "### Step 2: Isolate the term with ( d )", "Subtract 50 from both sides to eliminate the constant on the right:\n[\n150 - 50 = 45d \quad \Rightarrow \quad 100 = 45d\n]", "---", "### Step 3: Solve for ( d )", "To isolate ( d ), divide both sides by 45:\n[\nd = \frac{100}{45}\n]", "Simplify the fraction by dividing numerator and denominator by 5:\n[\nd = \frac{20}{9}\n]", "---", "### Final Answer and Summary", "The solution to ( 150 = \frac{10}{2}(2 \ imes 5 + 9d) ) is\n[\n\boxed{d = \frac{20}{9}}\n]", "---", "### Why This Equation Matters", "This problem introduces or reinforces key algebraic techniques:\n- Simplifying rational expressions\n- Order of operations and distributive property\n- Isolating variables via inverse operations\n- Handling fractions in equations", "Mastering equations like this builds confidence in tackling more complex expressions and prepares learners for advanced math topics such as systems of equations, linear inequalities, and calculus.", "---", "### Pro Tip: Always Verify Your Solution", "Plug ( d = \frac{20}{9} ) back into the original equation:\n[\n\frac{10}{2}(2 \ imes 5 + 9 \ imes \frac{20}{9}) = 5(10 + 20) = 5 \ imes 30 = 150\n]\nIt checks! You’ve solved it correctly.", "---", "If you're studying algebra, remember: equations are puzzles waiting to be solved. With each step—simplifying, isolating variables, and verifying—you strengthen your logical thinking and problem-solving skills.\nKeep practicing, stay curious, and keep solving!"]