\[ (64p + 16q + 4r + s) - (27p + 9q + 3r + s) = 84 - 39 \]

["Understanding the Simplified Equation: (64p + 16q + 4r + s) – (27p + 9q + 3r + s) = 84 – 39", "When tackling algebraic expressions, simplification is key to uncovering clarity and solving equations effectively. In this article, we explore and solve the equation:", "[\n(64p + 16q + 4r + s) - (27p + 9q + 3r + s) = 84 - 39\n]", "---", "### Step 1: Simplify the Left-Hand Side (LHS)", "Start by simplifying the expression on the left:", "[\n(64p + 16q + 4r + s) - (27p + 9q + 3r + s)\n]", "Distribute the negative sign across the second set of parentheses:", "[\n64p + 16q + 4r + s - 27p - 9q - 3r - s\n]", "Now combine like terms:", "- ( p )-terms: ( 64p - 27p = 37p )\n- ( q )-terms: ( 16q - 9q = 7q )\n- ( r )-terms: ( 4r - 3r = 1r ) or simply ( r )\n- ( s )-terms: ( s - s = 0 )", "Thus, the simplified LHS is:", "[\n37p + 7q + r\n]", "---", "### Step 2: Simplify the Right-Hand Side (RHS)", "The right-hand side is straightforward:", "[\n84 - 39 = 45\n]", "---", "### Step 3: Write the Simplified Equation", "Now our equation becomes:", "[\n37p + 7q + r = 45\n]", "This is a linear equation with three variables ( p, q, r ) and constant terms.", "---", "### Step 4: Analyze the Equation", "The equation ( 37p + 7q + r = 45 ) represents a plane in three-dimensional space. Since there are more variables than independent equations, the solution set includes infinitely many real solutions depending on the choices of two variables and subsequent determination of the third.", "For instance, solving for ( r ):", "[\nr = 45 - 37p - 7q\n]", "This form is helpful for graphing or numerical substitution, allowing flexible assignment to ( p ) and ( q ), then computing ( r ).", "---", "### Step 5: Real-World Context and Applications", "Such simplified equations often arise in optimization problems, resource allocation, or physics models where multiple variables interact linearly. By reducing expressions like ( (64p + 16q + 4r + s) - (27p + 9q + 3r + s) ), analysts efficiently evaluate performance differences or net changes.", "---", "### Conclusion", "The expression ( (64p + 16q + 4r + s) - (27p + 9q + 3r + s) = 84 - 39 ) simplifies elegantly to ( 37p + 7q + r = 45 ), offering a clear pathway for solving real-valued problems with multiple variables. By isolating one variable, practitioners gain a practical tool in algebra, engineering, and applied mathematics.", "---", "Keywords:\nalgebra simplification, linear equation, 64p + 16q + 4r + s – 27p + 9q + 3r + s = 84 – 39, solve for r, equation in three variables, plane equation, algebra solving tips, mathematical simplification", "---", "Further Reading:\n- How to Work with Linear Combinations in Algebra\n- Solving Systems of Linear Equations with Multiple Variables\n- Applications of Simplified Algebraic Expressions in Real Problems", "---", "By mastering expressions like this, you unlock deeper insight and efficiency in tackling complex mathematical challenges—whether in academics, programming, or engineering."]