["Understanding and Solving the Equation: 0.15x + 0.40(10 - x) = 2.5", "When solving equations involving real-world variables, equations like 0.15x + 0.40(10 - x) = 2.5 appear frequently in algebra, economics, and applied sciences. This article explains how to solve this linear equation step by step, interprets its meaning, and offers practical insights for students and professionals alike.", "---", "### Preparing to Solve the Equation", "We start with:", "$$
\n0.15x + 0.40(10 - x) = 2.5
\n$$", "This type of equation typically arises in contexts such as cost modeling, profit analysis, or weighted averages. Here, ( x ) represents an unknown quantity—a weight, a proportion, or a variableizing value—while the expression combines constants and ( x ) terms to balance an equation.", "---", "### Step 1: Expand the Parentheses", "Distribute the 0.40 across ( (10 - x) ):", "$$
\n0.15x + (0.40 \ imes 10) - (0.40 \ imes x) = 2.5
\n$$
\n$$
\n0.15x + 4 - 0.40x = 2.5
\n$$", "---", "### Step 2: Combine Like Terms", "Now combine the ( x )-terms:", "$$
\n(0.15 - 0.40)x + 4 = 2.5
\n$$
\n$$
\n-0.25x + 4 = 2.5
\n$$", "---", "### Step 3: Isolate the Variable", "Subtract 4 from both sides:", "$$
\n-0.25x = 2.5 - 4
\n$$
\n$$
\n-0.25x = -1.5
\n$$", "Now divide both sides by (-0.25):", "$$
\nx = \frac{-1.5}{-0.25} = \frac{1.5}{0.25} = 6
\n$$", "---", "### Verifying the Solution", "Plug ( x = 6 ) back into the original equation:", "$$
\n0.15(6) + 0.40(10 - 6) = 0.9 + 0.40 \ imes 4 = 0.9 + 1.6 = 2.5
\n$$", "The equation holds true, confirming ( x = 6 ) is correct.", "---", "### Interpreting the Result", "The solution ( x = 6 ) tells us that when ( x ) represents, for example, a portion of a total (such as 60% of 10 units), substituting ( x = 6 ) satisfies the relationship defined by the equation. This is useful for balancing budgets, determining percentage allocations, or solving physics and economics problems where parts contribute to a whole.", "---", "### Why This Equation Matters", "Linear equations like this are foundational in algebra and are models for real-life situations:", "- Budgeting: Distributing a fixed sum across categories with different rates
\n- Physics: Balancing forces or reconcile distances and times
\n- Business: Calculating break-even points or profit margins", "Understanding how to isolate variables and simplify expressions empowers you to model and solve equations across disciplines.", "---", "Summary
\nSolve ( 0.15x + 0.40(10 - x) = 2.5 ) by expanding, combining like terms, isolating ( x ), and verifying the result. This leads to ( x = 6 ), a critical value in many applied contexts.", "---", "Key Takeaways for Students and Professionals:
\n- Always simplify expressions step by step.
\n- Check work by substituting your answer back into the original equation.
\n- Recognize how algebraic models translate into practical decision-making.", "---", "Related Topics:
\n- Solving linear equations
\n- Distributive property in algebra
\n- Word problems and mathematical modeling
\n- Real-world applications of algebra", "---", "References:
\n- Algebra Fundamentals with Real-World Applications
\n- Khan Academy – Linear Equations
\n- College Algebra Textbook (e. g., Stewart, Redlin, Golub)", "---", "Master this equation type to build a strong algebraic foundation—essential for advanced study and practical problem-solving."]
