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

["# Solving the Factor Equation: ( y(x - 2) = x + 3 )", "When faced with the equation ( y(x - 2) = x + 3 ), solving for ( y ) or understanding its implications can unlock valuable insights in algebra and applied mathematics. This factor-based equation is a key example of how variables interact in linear relationships. In this article, we explore how to solve for ( y ), discuss its graphical interpretation, and highlight practical applications of this equation.", "## Understanding the Equation: ( y(x - 2) = x + 3 )", "The given equation, ( y(x - 2) = x + 3 ), is a linear equation involving two variables, ( x ) and ( y ). The term ( (x - 2) ) serves as a factor of ( y ), meaning it multiplies ( y ) to form the right-hand side, ( x + 3 ).", "To solve for ( y ), isolate it on one side of the equation:", "[\ny = \frac{x + 3}{x - 2}\n]", "This expression defines ( y ) explicitly in terms of ( x ), revealing that ( y ) is a rational function of ( x ).", "## Key Features of the Equation", "- Domain Restriction: The denominator ( x - 2 ) implies that ( x <br/>\neq 2 ), since division by zero is undefined. Thus, ( x = 2 ) is excluded from the domain.\n- Vertical Asymptote: At ( x = 2 ), the function ( y = \frac{x + 3}{x - 2} ) has a vertical asymptote. This vertical line is where the function becomes undefined and approaches infinity.\n- Horizontal Asymptote: As ( x \ o \pm\infty ), the function approaches ( y = 1 ), since leading terms dominate: ( \frac{x}{x} = 1 ).", "## Solving for Specific Values", "Suppose you want to evaluate ( y ) at a particular ( x ), say ( x = 4 ):", "[\ny(4 - 2) = 4 + 3 \implies 2y = 7 \implies y = 3.5\n]", "Alternatively, substitute ( x = 0 ):", "[\ny(0 - 2) = 0 + 3 \implies -2y = 3 \implies y = -\frac{3}{2}\n]", "These computations show how plugging values into the factored form enables quick evaluation.", "## Graphical Representation", "The function ( y = \frac{x + 3}{x - 2} ) can be graphed using key features:", "- Plot key points, including the intercepts and the vertical asymptote at ( x = 2 ).\n- Map the horizontal asymptote at ( y = 1 ).\n- Observe behavior near ( x = 2 ): as ( x \ o 2^+ ), ( y \ o +\infty ); as ( x \ o 2^- ), ( y \ o -\infty ).", "Graphing tools or manual plotting illustrate the hyperbolic shape characteristic of rational functions.", "## Applications and Real-World Context", "Equations in factor form such as ( y(x - 2) = x + 3 ) appear frequently in modeling:", "- Rate Problems: Relating variables in dynamic systems where ratios govern behavior.\n- Economic Models: Expressing supply or cost functions with constraints.\n- Physics: Describing relationships in motion and force where ratios are natural.", "Understanding how to manipulate and interpret such equations supports problem-solving across STEM disciplines.", "## Conclusion", "The equation ( y(x - 2) = x + 3 ) exemplifies how factoring reveals functional relationships in algebra. Solving for ( y ) yields ( y = \frac{x + 3}{x - 2} ), with important considerations around domain and asymptotes. Whether computing specific values, visualizing graphs, or applying the function in real-world scenarios, mastering this form enhances mathematical fluency and analytical problem-solving skills.", "If you found this article helpful, share it with fellow learners and explore more on rational functions and equation solving techniques!", "---", "Keywords: factor equation, solve for ( y ), ( y(x - 2) = x + 3 ), rational functions, algebraic solving, asymptotes, graphical analysis."]