Substituting the given values, \( x^2 + y^2 = 10^2 - 2 \times 21 = 100 - 42 = 58 \).

["# Mastering the Substitution of Values in Algebra: A Deep Dive into ( x^2 + y^2 = 10^2 - 2 \ imes 21 = 58 )", "In algebra, substitution is a powerful technique that allows you to simplify equations by replacing variables with specific values. Understanding how to correctly substitute values not only helps in solving complex equations but also forms the foundation for advanced problem solving in math, programming, and scientific modeling. This article explores the meaningful substitution task illustrated by the equation:", "[ x^2 + y^2 = 10^2 - 2 \ imes 21 = 100 - 42 = 58 ]", "## The Equation Breakdown", "At first glance, the equation", "[ x^2 + y^2 = 10^2 - 2 \ imes 21 ]", "might appear abstract, but its true value emerges when we compute the right-hand side step by step:", "- First, calculate ( 10^2 = 100 )\n- Then compute ( 2 \ imes 21 = 42 )\n- Finally subtract: ( 100 - 42 = 58 )", "Thus, the equation simplifies elegantly to:", "[\nx^2 + y^2 = 58\n]", "This is a standard equation representing a circle with radius ( \sqrt{58} ) centered at the origin in the coordinate plane. However, the real challenge lies in substituting specific values to explore relationships or solve for one variable in terms of the other.", "---", "## Why Substitute? The Role of Variable Assignment", "Substituting specific values into symbolic expressions transforms abstract algebra into a concrete exercise. For instance, choosing values for ( x ) or ( y ) enables:", "- Solving for one variable\n- Analyzing Pythagorean-like relationships\n- Testing symmetry or special cases\n- Gaining insight into geometric interpretations", "Given the equation simplifies to ( x^2 + y^2 = 58 ), let’s explore what happens when you substitute common values like ( x = 3 ), ( x = 4 ), or ( y = \sqrt{58 - x^2} ).", "---", "## Practical Examples of Substitution", "### Example 1: Fixing ( x = 3 )", "Set ( x = 3 ), then solve for ( y ):", "[\n3^2 + y^2 = 58 \Rightarrow 9 + y^2 = 58 \Rightarrow y^2 = 49 \Rightarrow y = \pm 7\n]", "So, two solutions are:\n- ( (3, 7) )\n- ( (3, -7) )", "Interpretation: These values show lattice points lying on the circle ( x^2 + y^2 = 58 ).", "---", "### Example 2: Fixing ( y = 5 )", "Set ( y = 5 ):", "[\nx^2 + 5^2 = 58 \Rightarrow x^2 = 58 - 25 = 33 \Rightarrow x = \pm \sqrt{33}\n]", "Solutions:\n- ( (\sqrt{33}, 5) )\n- ( (-\sqrt{33}, 5) )", "---", "### Example 3: Expressing ( y ) in terms of ( x )", "From ( x^2 + y^2 = 58 ), isolate ( y ):", "[\ny = \pm \sqrt{58 - x^2}\n]", "This formula lets anyone substitute any real number ( x ) within ( -\sqrt{58} \leq x \leq \sqrt{58} ) to find corresponding ( y )-values.", "---", "## Applications Beyond Algebra", "Substituting values like in this equation extends far beyond classroom exercises:", "- Physics: Calibrating models involving energy, motion, or forces often requires plugging real-world measurements.\n- Computer Science: Algorithms rely on substitution to test conditions or optimize loops.\n- Finance: Simulations use substituted variables to estimate outcomes under different scenarios.\n- Engineering: Circuit analysis and structural modeling employ similar algebraic manipulations.", "---", "## Summary", "The expression ( x^2 + y^2 = 10^2 - 2 \ imes 21 = 58 ) is more than a placeholder—it's a gateway to exploring computations, solutions, and relationships between variables. Substituting specific values illuminates concrete outcomes, validates theoretical results, and connects symbolic algebra to real numerical answers. Whether you're a student tackling equations, a developer implementing math logic, or a scientist modeling phenomena, mastering substitution techniques enhances precision and problem-solving agility.", "---", "### Further Reading", "- Explore the geometric meaning of ( x^2 + y^2 = r^2 )\n- Learn how substitution powers function solving\n- Study root isolation techniques and their application", "---", "Keywords: substitution in algebra, solving equations, ( x^2 + y^2 = 58 ), algebra practice, variable assignment, equation solving, coordinate geometry"]