["# Solving the Equation +0.50x = 0.35(3 + x): A Clear Step-by-Step Guide", "Understanding how to solve linear equations like +0.50x = 0.35(3 + x) is essential for mastering algebra. This equation appears often in math problem sets, science, economics, and engineering scenarios. In this article, we’ll break down the solution step-by-step, explain the logic behind each move, and provide insight into how such equations frequently appear in real-world contexts.", "## What is the Equation Setup?", "The equation in focus is:", "[
\n+0.50x = 0.35(3 + x)
\n]", "This equation involves one variable, x, on both sides, combined with constants and multiplication. Solving for x allows us to find the exact value that balances both sides.", "---", "## Step-by-Step Solution", "### Step 1: Expand the Right Side
\nMultiply the 0.35 through the parentheses:", "[
\n0.35(3 + x) = 0.35 \cdot 3 + 0.35 \cdot x = 1.05 + 0.35x
\n]", "Now the equation looks like:", "[
\n0.50x = 1.05 + 0.35x
\n]", "### Step 2: Eliminate Variables on One Side
\nSubtract 0.35x from both sides to isolate like terms:", "[
\n0.50x - 0.35x = 1.05
\n\Rightarrow 0.15x = 1.05
\n]", "### Step 3: Solve for x
\nDivide both sides by 0.15:", "[
\nx = \frac{1.05}{0.15} = 7
\n]", "---", "## Final Answer", "[
\n\boxed{x = 7}
\n]", "Verifying by plugging x = 7 back into the original equation confirms the solution:", "[
\n0.50(7) = 0.35(3 + 7) \Rightarrow 3.5 = 0.35 \cdot 10 = 3.5
\n]", "True on both sides — equation balanced!", "---", "## Real-World Contexts for Equations Like This", "Equations combining proportional relationships, such as +0.50x = k(3 + x), commonly appear in:", "- Financial calculations: Comparing growth rates, interest rates, or revenue models.
\n- Physics and engineering: Modeling motion, force, or electrical circuits.
\n- Economics: Analyzing cost, revenue, and profit relationships with variable inputs.
\n- Data science: Fitting linear regression lines or solving predictive formulas.", "Understanding how to manipulate and solve these equations empowers you to tackle a broad range of analytical problems.", "---", "## Summary", "- We began with +0.50x = 0.35(3 + x).
\n- Expanded the right-hand side using the distributive property.
\n- Collected x terms on one side:
\n [
\n 0.50x - 0.35x = 1.05 \Rightarrow 0.15x = 1.05
\n ]
\n- Solved for x to get x = 7.
\n- Verified the solution confirms correctness.", "Mastering this process builds a strong foundation for solving more complex equations and interpreting mathematical models across disciplines.", "For further study, explore similar equations involving fractions, decimals, or multiple variables—these enhance your algebraic fluency and problem-solving versatility.", "---", "Keywords: linear equations, solving for x, algebra tutorial, equation solution, +0.50x, 0.35(3 + x), mathematical problem solving, real-world math applications, step-by-step algebra."]
