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

["# Expand and Solve: A Complete Guide to the Equation ( xy - x = 2y + 3 )", "Solving equations step-by-step is a fundamental skill in algebra, especially when working with nonlinear expressions like ( xy - x = 2y + 3 ). This article explains how to expand, simplify, and solve this equation clearly and effectively. Whether you're a student learning algebra or a teacher guiding students, understanding the full process helps build strong problem-solving skills.", "## Understanding the Equation", "The given equation is:\n[\nxy - x = 2y + 3\n]", "This equation contains a product of variables ( xy ) and linear terms in ( x ) and ( y ). While it resembles a multiplicative relationship, our goal is to expand and transform it into a form that can be rearranged to isolate variables—ideal for solving for one variable in terms of the other.", "## Step 1: Expand Expressions to Standard Form", "Begin by expanding all terms:", "The left-hand side (LHS) ( xy - x ) is already expanded:\n[\nxy - x\n]", "The right-hand side (RHS) ( 2y + 3 ) is already fully expanded—no further expansion needed.", "So, the equation becomes:\n[\nxy - x = 2y + 3\n]", "Though already expanded, this form sets the stage for isolating variables. Since ( xy ) introduces a product, the next steps involve rearranging into standard linear or solvable forms.", "## Step 2: Rearranging Terms to Standard Algebraic Form", "To solve for one variable, it’s helpful to collect like terms. Move all terms involving ( x ) and ( y ) to one side and constants to the other:", "[\nxy - x - 2y = 3\n]", "Factor where possible:", "Notice ( x ) appears in both ( xy ) and ( -x ), so factor ( x ) from the first two terms:\n[\nx(y - 1) - 2y = 3\n]", "Now the equation is:\n[\nx(y - 1) - 2y = 3\n]", "This is a simplified linear form, isolating ( x ):", "## Step 3: Solve for ( x ) in Terms of ( y )", "Start by isolating ( x(y - 1) ):\n[\nx(y - 1) = 3 + 2y\n]", "Now divide both sides by ( y - 1 ), assuming ( y <br/>\ne 1 ):\n[\nx = \frac{3 + 2y}{y - 1}\n]", "This expression gives ( x ) explicitly in terms of ( y )—a key insight for solving the equation.", "## Optional: Express ( y ) in Terms of ( x )", "To fully solve, rearrange the original equation to solve for ( y ):", "Starting over:\n[\nxy - x = 2y + 3\n]", "Move all terms with ( y ) to the left:\n[\nxy - 2y = x + 3\n]", "Factor ( y ) on the left:\n[\ny(x - 2) = x + 3\n]", "Now divide both sides by ( x - 2 ), assuming ( x <br/>\ne 2 ):\n[\ny = \frac{x + 3}{x - 2}\n]", "---", "## Practical Applications and Summary", "Solving equations like ( xy - x = 2y + 3 ) helps in various real-world contexts, including:", "- Analyzing linear relationships in economics\n- Modeling rates and ratios in physics\n- Simplifying algebraic expressions in higher math", "### Key Steps Recap:\n- Expand and simplify: ( xy - x = 2y + 3 )\n- Rearrange into standard form: ( xy - 2y - x = 3 )\n- Factor and isolate variables: ( x(y - 1) = 2y + 3 \Rightarrow x = \frac{2y + 3}{y - 1} )\n- Also, solve for ( y ): ( y = \frac{x + 3}{x - 2} ) (valid when ( x <br/>\ne 2 ))", "Understanding these steps builds a solid foundation for tackling more complex equations and systems in algebra.", "---", "## Final Thoughts", "The equation ( xy - x = 2y + 3 ) may look simple but illustrates important algebraic techniques: expanding expressions, rearranging terms, and isolating variables. Mastering expansion and transformation is crucial for problem-solving in mathematics. With consistent practice, expanding and solving such equations becomes intuitive and powerful.", "---", "Keywords: expand equation ( xy - x = 2y + 3 ), solve for ( x ) in terms of ( y ), solve for ( y ) in terms of ( x ), algebraic manipulation, linear equations, product terms, factoring, solving algebraic expressions.\nTags: algebra tutorial, solving equations, expand equations guide, linear algebra, step-by-step solving, intermediate algebra.", "---\nExplore related articles:\n- How to Solve Nonlinear Equations: Tips and Strategies\n- Mastering Variable Isolation in Algebra\n- From Expansion to Standard Form: A Step-by-Step Approach", "---", "Author: Math Tutoring Team | Last Updated: April 2025"]