Solve \( 2x + 3y = 120 \) for \( x = 30, y = 20 \): \( R = 50(30) + 80(20) = 1500 + 1600 = 3100 \)

["Understanding the Equation: Solving ( 2x + 3y = 120 ) with Example Values ( x = 30 ) and ( y = 20 )", "Solving linear equations is a fundamental skill in algebra, useful in fields ranging from economics to engineering. One such equation is ( 2x + 3y = 120 ), a classic example of a two-variable linear relationship. In this article, we’ll explore how to solve this equation, verify specific values, and interpret the result with a detailed calculation.", "---", "### What is the Equation ( 2x + 3y = 120 )?", "The equation ( 2x + 3y = 120 ) represents a straight line when graphed on a coordinate plane. It relates two variables — ( x ) and ( y ) — such that for any pair ((x, y)) satisfying this equation, the linear combination ( 2x + 3y ) always equals 120.", "This form is commonly used to model practical problems, such as budgeting, resource allocation, or constraints in optimization.", "---", "### How to Solve for ( x ) or ( y )", "While the equation defines a relationship between ( x ) and ( y ), we often solve for one variable in terms of the other.", "Solve for ( x ):\n[ 2x = 120 - 3y ]\n[ x = \frac{120 - 3y}{2} ]", "Solve for ( y ):\n[ 3y = 120 - 2x ]\n[ y = \frac{120 - 2x}{3} ]", "These formulas allow us to express one variable depending on the other — essential for substitution or graphical solution methods.", "---", "### Example Values: ( x = 30 ) and ( y = 20 )", "Let’s verify that ( x = 30 ) and ( y = 20 ) satisfy the original equation:", "[\n2(30) + 3(20) = 60 + 60 = 120\n]", "The left side equals the right side, confirming the values are consistent with the equation.", "---", "### Calculate the Expression ( R = 50x + 80y )", "In many applied problems, expressions like ( R = 50x + 80y ) represent totals — perhaps revenue from selling items priced at $50 and $80 respectively, with ( x ) and ( y ) being counts.", "Using ( x = 30 ) and ( y = 20 ):", "[\nR = 50(30) + 80(20) = 1500 + 1600 = 3100\n]", "Thus, when 30 units at $50 and 20 units at $80 are sold, the total ( R = $3100 ).", "---", "### Why This Matters: Real-World Application", "The method demonstrated here — verifying solution pairs and computing weighted totals — applies broadly in finance, operations research, and data modeling. For example:", "- Budget Optimization: Choosing quantities ( x ) and ( y ) (products) subject to budget constraints ( 2x + 3y \leq 120 )\n- Revenue Projection: Estimating total sales using per-unit prices and quantities\n- Resource Mixing: Calculating totals from combined inputs where each input has different cost or value", "---", "### Summary", "Solving ( 2x + 3y = 120 ) yields a family of solutions representing all point pairs ((x, y)) that satisfy the constraint. Substituting example values ( x = 30 ), ( y = 20 ) verifies correctness while calculating the expression ( R = 50x + 80y ) gives a clear, meaningful total. Mastery of such algebraic relationships underpins more advanced mathematical and analytical techniques.", "---", "Keywords: Solve linear equation, ( 2x + 3y = 120 ), verify ( x = 30, y = 20 ), R = 50x + 80y, linear combination, algebra example, applied math, constraint solving."]