\[ yx - y = 2x + 3 \] - United Radiology

["# Solving the Equation ( yx - y = 2x + 3 ): A Complete Guide", "If you're working on solving equations involving variables multiplied together, such as ( yx - y = 2x + 3 ), you're not alone. This type of equation—where variables appear both added and multiplied—challenges many learners. In this article, we’ll break down the step-by-step solution process, explain key concepts, and explore how to work with expressions containing mixed variable terms. Whether you're a high school student, a math enthusiast, or someone needing a clear review, this guide provides insights into solving equations of the form ( yx - y = 2x + 3 ).", "---", "## Understanding the Equation: ( yx - y = 2x + 3 )", "At first glance, ( yx - y = 2x + 3 ) may look complex because it combines:", "- Multiplication: ( yx ) (same as ( xy ))
\n- Subtraction: ( yx - y )
\n- Linear terms: ( 2x + 3 )", "This equation is linear in both ( x ) and ( y ), but the presence of ( yx ) means it’s not standard linear. Our goal is to isolate variables and solve for one in terms of the other.", "---", "## Step 1: Factor where possible", "Notice that ( yx - y ) shares a common factor ( y ). Factor it out:", "[
\nyx - y = y(x - 1)
\n]", "So the original equation becomes:", "[
\ny(x - 1) = 2x + 3
\n]", "---", "## Step 2: Solve for ( y )", "To isolate ( y ), divide both sides by ( x - 1 ), assuming ( x <br/>\ne 1 ):", "[
\ny = \frac{2x + 3}{x - 1}
\n]", "This is the explicit solution:
\n( y = \frac{2x + 3}{x - 1} )", "---", "## Step 3: Domain Considerations", "Important: The denominator ( x - 1 ) cannot be zero, so:", "[
\nx <br/>\ne 1
\n]", "When ( x = 1 ), the original equation leads to division by zero, which is undefined. Therefore, ( x = 1 ) is excluded from the solution set.", "---", "## Step 4: Verification", "Let’s verify by plugging in a test value, say ( x = 2 ):", "Left side:
\n( yx - y = y(2) - y = 2y - y = y )", "With ( y = \frac{2(2)+3}{2-1} = \frac{7}{1} = 7 ), so left side = 7.", "Right side:
\n( 2x + 3 = 2(2) + 3 = 4 + 3 = 7 )", "Both sides match, confirming correctness.", "---", "## Step 5: Graphical Interpretation (Optional)", "The equation ( y = \frac{2x + 3}{x - 1} ) represents a rational function with a vertical asymptote at ( x = 1 ), and a slant asymptote found by polynomial long division:", "Divide ( 2x + 3 ) by ( x - 1 ):
\n[
\n\frac{2x + 3}{x - 1} = 2 + \frac{5}{x - 1}
\n]", "So asymptote: ( y = 2 ), excluding ( x = 1 ).", "---", "## Final Thoughts", "Solving equations like ( yx - y = 2x + 3 ) involves:", "- Factoring to simplify
\n- Recognizing domain restrictions
\n- Isolating variables using algebra
\n- Verifying through substitution", "Understanding such equations strengthens algebra skills essential for higher math, including calculus, differential equations, and applied modeling.", "---", "## Key Takeaways", "- Factoring transforms mixed terms into simpler expressions.
\n- Always check for undefined values (like division by zero).
\n- The solution ( y = \frac{2x + 3}{x - 1} ) is valid for all ( x <br/>\ne 1 ).
\n- Graphing reveals critical behavior near discontinuities and asymptotes.", "---", "Keywords: ( yx - y = 2x + 3 ), solve linear equations, factoring algebra, solving rational equations, vertical asymptote, domain restrictions, algebra tips, step-by-step solution.", "---", "If you're looking to master equation-solving techniques, practice similar problems with mixed variable products and explore how factoring and substitution simplify solutions. Understanding ( yx - y = 2x + 3 ) is a foundational step toward more advanced algebraic reasoning!"]