\[ y(x - 1) = 2x + 3 \] - United Radiology

["# Solving the Equation ( y(x - 1) = 2x + 3 ): A Step-by-Step Guide", "If you've ever encountered an equation like ( y(x - 1) = 2x + 3 ), solving for ( y ) is straightforward—but understanding how to manipulate and interpret such equations is key in algebra and beyond. This article walks you through solving ( y(x - 1) = 2x + 3 ) with clear steps, tips, and practical applications.", "## Understanding the Equation", "Equation form:
\n[ y(x - 1) = 2x + 3 ]", "Here, ( y ) is expressed in terms of ( x ), and your goal is to isolate ( y ).", "---", "## Step 1: Isolate ( y )", "Start by dividing both sides of the equation by ( (x - 1) ), assuming ( x <br/>\ne 1 ) (since division by zero is undefined):", "[
\ny = \frac{2x + 3}{x - 1}
\n]", "This expression represents ( y ) as a rational function of ( x ).", "---", "## Step 2: Simplify (if possible)", "The right-hand side is already in simplest form. However, if needed, you can expand the numerator and keep the structure:", "[
\ny = \frac{2x + 3}{x - 1}
\n]", "This form helps interpret the function’s behavior such as asymptotes and domain restrictions.", "---", "## Step 3: Domain Consideration", "Recall that ( x = 1 ) makes the denominator zero, so ( x = 1 ) is excluded from the domain. Thus, the solution is valid for all ( x <br/>\ne 1 ).", "---", "## Step 4: Interpret the Result", "The equation ( y = \frac{2x + 3}{x - 1} ) describes a hyperbola with a vertical asymptote at ( x = 1 ). Understanding this helps when graphing or analyzing function behavior.", "---", "## Real-World Applications", "Equations like ( y(x - 1) = 2x + 3 ) appear in physics (relationships between variables), economics (cost and revenue models), and engineering (behavioral equations). Learning to solve such equations empowers problem-solving across disciplines.", "---", "## Summary", "- Start with ( y(x - 1) = 2x + 3 )
\n- Divide both sides by ( x - 1 ) (for ( x <br/>\ne 1 ))
\n- Simplify to ( y = \frac{2x + 3}{x - 1} )
\n- Note domain restriction: ( x <br/>\ne 1 )
\n- Apply knowledge in graphs, calculus, or applied modeling", "---", "## Further Reading", "- How to solve linear equations in one variable
\n- Rational functions and asymptotes
\n- Applications of algebraic equations in real science and engineering", "---", "Keywords: solve ( y(x - 1) = 2x + 3 ), isolate ( y ), rational function, algebra tutorial, solve linear equations, domain restriction, mathematical method, hyperbola function.", "---", "By mastering this simple yet essential equation, you build a strong foundation for tackling more complex algebraic and functional problems."]