Rearrange: \( xy - 2y = x + 3 \).

["Title: How to Rearrange the Equation ( xy - 2y = x + 3 ) — Step-by-Step Guide", "Meta Description:\nLearn how to rearrange the equation ( xy - 2y = x + 3 ) into standard forms! This comprehensive guide explains algebraic manipulation, isolating variables, and solving for specific terms. Boost your algebra skills today!", "---", "### Understanding the Equation: ( xy - 2y = x + 3 )", "The equation ( xy - 2y = x + 3 ) is a linear equation involving two variables, ( x ) and ( y ). Rearranging it helps simplify solving for one variable in terms of the other and supports graphing or substitution methods. In this article, we’ll break down how to rearrange this equation and explore key algebraic techniques.", "---", "### Step 1: Collect Like Terms", "Start by moving all terms to one side to group like variables. Subtract ( x ) and ( 3 ) from both sides:", "[\nxy - 2y - x - 3 = 0\n]", "This puts the equation in standard form with all variable and constant terms on one side:", "[\nxy - 2y - x - 3 = 0\n]", "---", "### Step 2: Factor Where Possible", "To rearrange for specific variables, factor common terms. Notice that ( y ) appears in the first two terms:", "Factor ( y ) out:", "[\ny(x - 2) - x - 3 = 0\n]", "Now move the remaining terms ( -x - 3 ) to the right-hand side:", "[\ny(x - 2) = x + 3\n]", "---", "### Step 3: Solve for a Variable (if needed)", "- Solve for ( y ):\n Divide both sides by ( (x - 2) ) (assuming ( x <br/>\ne 2 )):", "[\n y = \frac{x + 3}{x - 2}\n ]", "- Solve for ( x ):\n First isolate terms with ( x ). Start again from ( y(x - 2) = x + 3 ), expand:", "[\n xy - 2y = x + 3\n ]", "Bring all ( x )-terms to one side:", "[\n xy - x = 2y + 3\n ]", "Factor ( x ) on the left:", "[\n x(y - 1) = 2y + 3\n ]", "Solve for ( x ):", "[\n x = \frac{2y + 3}{y - 1}, \quad \ ext{for } y <br/>\ne 1\n ]", "---", "### Step 4: Rearranged Form for Graphing or Substitution", "The equation can be written as:", "[\ny(x - 2) = x + 3\n]", "Or after solving for ( y ):", "[\ny = \frac{x + 3}{x - 2}\n]", "This linear-rational form is ideal for graphing (asymptotes at ( x = 2 )), plugging values, or substitution in systems.", "---", "### Why Rearranging This Equation Matters", "- Clear Analysis: Isolating variables makes relationships transparent.\n- Easier Solving: Rearranged forms support substitution and elimination methods.\n- Graphical Insight: Expressions like ( y = \frac{x + 3}{x - 2} ) reveal hyperbola behavior.\n- Real-World Applications: In physics, economics, or engineering, rearranged equations often reflect constraints or revertible processes.", "---", "### Summary", "To rearrange ( xy - 2y = x + 3 ):\n1. Move all terms to one side: ( xy - 2y - x - 3 = 0 )\n2. Factor common variables to isolate ( y ) or ( x )\n3. Simplify algebraically, respecting domain restrictions\n4. Express in simplified forms for solving or graphing", "---", "### FAQ: Common Questions About Rearranging ( xy - 2y = x + 3 )", "Q: Can I rearrange the equation differently?\nA: Yes! Alternate forms like ( x(y - 1) = 2y + 3 ) are equally valid and useful depending on your goal.", "Q: What if ( x = 2 ) or ( y = 1 )?\nA: These make denominators zero in solved forms. Avoid these values as they cause undefined expressions.", "Q: How does this equation look after graphing?\nA: Rewriting as ( y = \frac{x + 3}{x - 2} ) highlights a rational function with a vertical asymptote at ( x = 2 ).", "---", "### Final Tips", "- Always check for restrictions (like divisions by zero).\n- Practice with substituting values for ( x ) or ( y ) to verify consistency.\n- Use rearranged forms for solving systems or modeling real-world scenarios.", "---", "Keywords: rearrange ( xy - 2y = x + 3 ), algebraic manipulation, solve for y, solve for x, factor variables, linear equation rearrangement, solving rational equations, step-by-step algebra, coordinate geometry, equation simplification.", "---", "Optimizing your approach to rearranging equations like ( xy - 2y = x + 3 ) enhances your algebraic fluency and opens doors to advanced mathematics and practical problem-solving. Start practicing today!"]