["# How to Solve the Equation ( 10 = 0.10(200 - x) ) – Step-by-Step Guide", "Solving linear equations is a foundational skill in algebra, widely used in math, science, engineering, and everyday problem-solving. One common type of equation students encounter is one involving decimals, such as ( 10 = 0.10(200 - x) ). In this article, we’ll break down how to solve this equation and explore its real-world applications.", "---", "## What Is the Equation?", "The equation we aim to solve is:", "[
\n10 = 0.10(200 - x)
\n]", "This form commonly appears in problems involving proportions, percentages, or cost calculations. Understanding how to isolate the variable ( x ) is key to finding its value.", "---", "## Step-by-Step Solution", "### Step 1: Expand the right side
\nStart by distributing ( 0.10 ) across the parentheses:", "[
\n10 = 0.10 \ imes 200 - 0.10 \ imes x
\n]
\n[
\n10 = 20 - 0.10x
\n]", "### Step 2: Isolate the variable term
\nSubtract 20 from both sides to move the constant to the left:", "[
\n10 - 20 = -0.10x
\n]
\n[
\n-10 = -0.10x
\n]", "### Step 3: Solve for ( x )
\nDivide both sides by ( -0.10 ) (equivalent to multiplying by ( -10 )):", "[
\n\frac{-10}{-0.10} = x
\n]
\n[
\nx = 100
\n]", "---", "## Final Answer", "[
\nx = 100
\n]", "This means that when ( x = 100 ), the original equation holds true:", "[
\n10 = 0.10(200 - 100) = 0.10 \ imes 100 = 10
\n]", "---", "## Real-World Applications", "Equations like ( 10 = 0.10(200 - x) ) model scenarios such as:", "- Discount calculations: If a 10% discount on $200 minus an amount ( x ) equals $10, solving shows that $100 was the discounted portion.
\n- Distance and speed problems: Estimating travel distances under different conditions.
\n- Budgeting and finance: Analyzing expenses where a fixed amount is allocated or reduced by a fraction.", "---", "## Tips for Mastering This Type of Equation", "- Always distribute decimals carefully to avoid sign errors.
\n- Keep track of signs when dividing by negative numbers.
\n- Reverse order of operations applies: undo multiplication/division before addition/subtraction.
\n- Always verify your solution by plugging ( x = 100 ) back into the original equation.", "---", "## Conclusion", "Solving ( 10 = 0.10(200 - x) ) teaches essential algebraic techniques that extend beyond this equation—it strengthens your ability to model, analyze, and solve real-world problems. With practice, decoding equations becomes intuitive and powerful.", "If you found this guide helpful, explore our other posts on algebraic techniques and word problem solving to sharpen your math skills.", "---", "### Key SEO Keywords:
\nsolve equation 10 = 0.10(200 - x), linear equation solution, algebra step-by-step, how to solve decimal equations, real-world math problems, algebra practice problems, solving linear equations with decimals.", "---", "Keywords omitted for readability; focus remains on clear explanation and practical value."]
