["Solving (5x - 4) / (7x + 2) = 1 / 2: A Step-by-Step Guide to Finding the Correct Value of x", "If you’re tackling the equation (5x - 4) / (7x + 2) = 1 / 2, you're in the right place. This article will guide you through each step to solve the equation, factor in the important “after changes” perspective, and help you understand key algebraic techniques. Whether you're a student, math enthusiast, or professional, this breakdown will enhance your problem-solving skills for rational equations.", "---", "### Understanding the Equation: What’s the “After Change”?", "The phrase “after changes” can be interpreted as solving the equation after simplification—those moment-to-moment transformations that reveal the true value of the unknown variable. Here, we’ll walk through solving for x, the unknown, given this rational equation.", "---", "### Step 1: Set Up the Equation", "We start with:
\n[
\n\frac{5x - 4}{7x + 2} = \frac{1}{2}
\n]", "This equation states that a fraction with numerator (5x – 4) and denominator (7x + 2) is equal to ½. Transforming this equation reveals how changing values for x affects balance—this “after change” logic allows us to isolate x.", "---", "### Step 2: Cross-Multiply to Eliminate the Fork", "Cross-multiplying clears the fraction:", "[
\n2(5x - 4) = 1(7x + 2)
\n]", "Expanding both sides:", "[
\n10x - 8 = 7x + 2
\n]", "This step simplifies the equation—one of the key “after changes” transformations that exposes linear behavior beneath the rational form.", "---", "### Step 3: Move All Terms to One Side", "Subtract 7x and add 8 to both sides:", "[
\n10x - 7x = 2 + 8
\n]", "[
\n3x = 10
\n]", "Here, x “shifts” into focus—now solvable. This move exemplifies the “after change” where algebraic manipulation reveals clarity.", "---", "### Step 4: Isolate x", "Divide both sides by 3:", "[
\nx = \frac{10}{3}
\n]", "---", "### Step 5: Verify the Solution (Critical Step!)", "Substitute x = 10/3 back into the original equation to validate:", "Numerator:
\n[
\n5\left(\frac{10}{3}\right) - 4 = \frac{50}{3} - \frac{12}{3} = \frac{38}{3}
\n]", "Denominator:
\n[
\n7\left(\frac{10}{3}\right) + 2 = \frac{70}{3} + \frac{6}{3} = \frac{76}{3}
\n]", "Fraction:
\n[
\n\frac{38/3}{76/3} = \frac{38}{3} \cdot \frac{3}{76} = \frac{38}{76} = \frac{1}{2}
\n]", "✅ The left side equals the right—confirming the solution.", "---", "### Final Thoughts: Why This Matters", "Solving (5x - 4)/(7x + 2) = 1/2 isn’t just about math—it’s about understanding transformation logic in equations, how simplification “after changes” uncovers unknowns, and verifying each step’s accuracy. Always test your solution to avoid mathematical detours.", "---", "### Want More? Try These Variations!", "- Solve (3x - 1)/(2x + 5) = 2
\n- Explore how domain restrictions affect the solution
\n- Practice rational equations with changing coefficients", "---", "### Summary", "Final value:
\n[
\n\boxed{x = \frac{10}{3}}
\n]
\nAfter carefully applying algebraic transformations, verifying, and confirming, your solution is accurately found.", "---", "Keywords:
\n(5x - 4) / (7x + 2) = 1/2, solve rational equations, algebraic steps, after changes in equations, step-by-step solution, verify x, solving for x algebraically, intermediate algebra", "---", "Ready to tackle your next equation? Use this method to drive transformation, understand change, and find exact solutions."]
