\[ (8p + 4q + 2r + s) - (p + q + r + s) = 14 - 3 \]

["Understanding the Equation: (8p + 4q + 2r + s) – (p + q + r + s) = 14 – 3", "When working with algebraic expressions, simplifying and interpreting equations is key to grasping their meaning and applications—especially in fields like economics, engineering, or optimization. One such expression you may encounter is:", "(8p + 4q + 2r + s) – (p + q + r + s) = 14 – 3", "Let’s break this down step-by-step to solve and understand the equality.", "---", "### Step 1: Simplify the Left Side", "Start by distributing the subtraction across the second parentheses:", "[\n(8p + 4q + 2r + s) - (p + q + r + s) = 8p + 4q + 2r + s - p - q - r - s\n]", "Now combine like terms:", "- For ( p ): ( 8p - p = 7p )\n- For ( q ): ( 4q - q = 3q )\n- For ( r ): ( 2r - r = r )\n- For ( s ): ( s - s = 0 )", "So the left side simplifies to:", "[\n7p + 3q + r\n]", "---", "### Step 2: Simplify the Right Side", "The right side is:", "[\n14 - 3 = 11\n]", "---", "### Step 3: Set the Simplified Forms Equal", "Now our equation becomes:", "[\n7p + 3q + r = 11\n]", "This is a linear Diophantine equation in three variables (( p, q, r )) equal to a constant. It defines a plane in three-dimensional space, where all ordered triples ((p, q, r)) satisfying this equation lie.", "---", "### Step 4: Interpreting the Equation", "This equation represents a relationship among variables typically used in optimization models, substitution methods, or systems of equations. For example, if ( s ) does not appear in the final simplified form, it is a dependent variable in this equation—its value can be expressed in terms of ( p, q, ) and ( r ), provided a relationship exists.", "---", "### Step 5: Solving for One Variable (Optional)", "Suppose you want to express ( r ) in terms of ( p ) and ( q ):", "[\nr = 11 - 7p - 3q\n]", "This form helps analyze how changes in ( p ) and ( q ) affect ( r ), useful in sensitivity analysis, linear programming, or differential systems.", "---", "### Step 6: Practical Applications", "Expressions like this arise in budget modeling, resource allocation problems, and multivariable calculus. By reducing complexity and isolating key variables, such equations aid in decision-making, simulation, and theoretical analysis.", "---", "### Conclusion", "The equation\n[\n(8p + 4q + 2r + s) - (p + q + r + s) = 14 - 3\n]\nsimplifies to\n[\n7p + 3q + r = 11\n]\na vital linear relation in algebra with broad applications. Understanding this facilitates effective modeling and problem-solving across STEM disciplines.", "---", "Keywords for SEO:\nalgebraic simplification, equation solving, linear equation 7p + 3q + r = 11, variable relationship, Diophantine equation, simplifying expressions, multivariable equation interpretation, applications of linear equations", "---", "Meta Description for Business/SEO use:\nLearn how to simplify the equation (8p + 4q + 2r + s) – (p + q + r + s) = 14 – 3 into 7p + 3q + r = 11 — a key step in solving linear systems, commonly used in optimization, modeling, and algebra."]