$ x(4 - y) + y(2 - x) = 4x - xy + 2y - xy = 4x + 2y - 2xy = -2 $.

["Optimize Your Equation: Solving $ x(4 - y) + y(2 - x) = 4x + 2y - 2xy = -2 $", "If you’ve ever faced a tricky algebraic equation that seems tangled at first glance, solving expressions like $ x(4 - y) + y(2 - x) = 4x + 2y - 2xy = -2 $ can feel overwhelming. But with a strategic approach, you can simplify, solve, and understand this equation with confidence.", "In this SEO-optimized guide, we’ll unpack the equation $ x(4 - y) + y(2 - x) = -2 $, simplify it into standard form, solve for one variable in terms of the other, and explore its implications in algebra and real-world applications.", "---", "### Step 1: Expand and Simplify the Equation", "Start by expanding each term in the expression:", "$$\nx(4 - y) + y(2 - x) = 4x - xy + 2y - xy\n$$", "Combine like terms:", "$$\n4x - xy + 2y - xy = 4x + 2y - 2xy\n$$", "So the equation becomes:", "$$\n4x + 2y - 2xy = -2\n$$", "---", "### Step 2: Rewrite in Standard Form", "To solve this equation, it helps to bring all terms to one side:", "$$\n4x + 2y - 2xy + 2 = 0\n$$", "We can rewrite it as:", "$$\n-2xy + 4x + 2y + 2 = 0\n$$", "For convenience, multiply both sides by $-1$ to make coefficients positive (this won’t change the solution set):", "$$\n2xy - 4x - 2y - 2 = 0\n$$", "---", "### Step 3: Solve for One Variable in Terms of the Other", "Let’s isolate $ y $. Start with:", "$$\n2xy - 4x - 2y - 2 = 0\n$$", "Group terms with $ y $:", "$$\n2xy - 2y = 4x + 2\n$$", "Factor $ y $ on the left:", "$$\ny(2x - 2) = 4x + 2\n$$", "Now, solve for $ y $:", "$$\ny = \frac{4x + 2}{2x - 2}\n$$", "We can factor numerator and denominator to simplify:", "$$\ny = \frac{2(2x + 1)}{2(x - 1)} = \frac{2x + 1}{x - 1}, \quad \ ext{for } x <br/>\ne 1\n$$", "---", "### Step 4: Analyze the Solution", "The simplified expression:", "$$\ny = \frac{2x + 1}{x - 1}, \quad x <br/>\ne 1\n$$", "is valid for all $ x <br/>\ne 1 $. When $ x = 1 $, the original denominator becomes zero, so the point is excluded.", "This equation defines a hyperbolic relationship between $ x $ and $ y $. It’s useful in optimization problems, curve modeling, or simplifying complex expressions in engineering and economics.", "Note: If substitution or graphing is required, this rational function enables plotting and identifying asymptotes (vertical at $ x = 1 $, horizontal as $ x \ o \pm\infty $).", "---", "### Step 5: Real-World Applications", "Equations like $ x(4 - y) + y(2 - x) = 4x + 2y - 2xy $ appear in:", "- Economics: Modeling supply-demand interactions where utility functions depend linearly on two variables.\n- Physics: Deriving equilibrium conditions in mechanical or thermodynamic systems.\n- Data Science: Simplifying interaction terms in regression models or machine learning feature engineering.", "Understanding how to simplify and solve such expressions gives clarity in modeling scenarios involving interdependent variables.", "---", "### Summary", "The equation:", "$$\nx(4 - y) + y(2 - x) = -2\n$$", "is algebraically simplified to:", "$$\n2xy - 4x - 2y - 2 = 0\n\quad \ ext{or} \quad\ny = \frac{2x + 1}{x - 1}, \quad x <br/>\ne 1\n$$", "This rational function gives a clear relationship between $ x $ and $ y $, useful across disciplines. Mastering step-by-step simplification and solution strategies helps unlock deeper insight in algebra and applied problem-solving.", "---", "Keywords: solve $ x(4 - y) + y(2 - x) = 4x + 2y - 2xy $, simplify $ 2xy - 4x - 2y - 2 = 0 $, solve for $ y $ in terms of $ x $, rational equation solutions, algebraic manipulation tips, real-world algebraic applications.", "Meta Description:\nLearn how to simplify and solve the equation $ x(4 - y) + y(2 - x) = -2 $ step-by-step. Discover the rational function solution $ y = \frac{2x + 1}{x - 1} $, ideal for algebra students and professionals modeling interdependent variables.", "---", "If you’re studying or teaching algebra and want a clear path to solving such expressions, mastering this equation will strengthen your foundation and confidence. Start simplifying today!"]