نعوض $z = x$ و $y = a - x$ في $x^2 + y^2 + z^2 = 2a^2$:

["SEO Optimized Article: Solving $ <br/>\nearrow z = x $ and $ y = a - x $ in the Equation $ x^2 + y^2 + z^2 = 2a^2 $", "---", "Understanding the Geometry Behind $ z = x $, $ y = a - x $, and the Curve $ x^2 + y^2 + z^2 = 2a^2 $", "In algebra and geometry, solving systems of equations often reveals hidden patterns and real-world relationships. One such elegant setup involves substituting linear expressions into a quadratic surface defined by $ x^2 + y^2 + z^2 = 2a^2 $. Here, we explore how replacing $ z $ with $ x $ and $ y $ with $ a - x $ transforms the equation into a simpler, solvable curve — and uncover the mathematical insights behind it.", "---", "### Step 1: Substitute Known Expressions", "We are given:", "[\nz = x \quad \ ext{and} \quad y = a - x\n]", "Substitute these into the equation:", "[\nx^2 + y^2 + z^2 = 2a^2\n]", "becomes:", "[\nx^2 + (a - x)^2 + x^2 = 2a^2\n]", "---", "### Step 2: Expand and Simplify", "First, expand $ (a - x)^2 $:", "[\n(a - x)^2 = a^2 - 2ax + x^2\n]", "Now substitute:", "[\nx^2 + (a^2 - 2ax + x^2) + x^2 = 2a^2\n]", "Simplify the left-hand side:", "[\nx^2 + a^2 - 2ax + x^2 + x^2 = 3x^2 - 2ax + a^2\n]", "Thus:", "[\n3x^2 - 2ax + a^2 = 2a^2\n]", "---", "### Step 3: Rearrange into a Standard Quadratic", "Bring all terms to one side:", "[\n3x^2 - 2ax + a^2 - 2a^2 = 0\n]", "[\n3x^2 - 2ax - a^2 = 0\n]", "This is a quadratic equation in $ x $:", "[\n3x^2 - 2a x - a^2 = 0\n]", "---", "### Step 4: Solve the Quadratic Equation", "Use the quadratic formula:", "[\nx = \frac{2a \pm \sqrt{(-2a)^2 - 4 \cdot 3 \cdot (-a^2)}}{2 \cdot 3}\n]", "Calculate the discriminant:", "[\n(−2a)^2 - 4(3)(−a^2) = 4a^2 + 12a^2 = 16a^2\n]", "So:", "[\nx = \frac{2a \pm \sqrt{16a^2}}{6} = \frac{2a \pm 4a}{6}\n]", "This yields two solutions:", "[\nx = \frac{2a + 4a}{6} = \frac{6a}{6} = a \quad \ ext{and} \quad x = \frac{2a - 4a}{6} = \frac{-2a}{6} = -\frac{a}{3}\n]", "---", "### Step 5: Find Corresponding $ y $ and $ z $ Values", "Recall: $ y = a - x $, $ z = x $", "First solution: $ x = a $\nThen:\n$ y = a - a = 0 $, $ z = a $", "→ Point: $ (a, 0, a) $", "Second solution: $ x = -\frac{a}{3} $\nThen:\n$ y = a - \left(-\frac{a}{3}\right) = a + \frac{a}{3} = \frac{4a}{3} $,\n$ z = -\frac{a}{3} $", "→ Point: $ \left(-\frac{a}{3}, \frac{4a}{3}, -\frac{a}{3}\right) $", "---", "### Step 6: Interpret the Geometric Meaning", "The curve formed by $ x^2 + y^2 + z^2 = 2a^2 $ is a sphere of radius $ \sqrt{2}a $ centered at the origin. The substitutions $ z = x $ and $ y = a - x $ define a hyperbolic paraboloid-like constraint plane, intersecting the sphere in a one-dimensional curve — a conic section — specifically, two points in this case (real solutions), indicating tangency or intersection along a line segment.", "This reflects how linear constraints reduce a 3D surface to a finite set of points — a common technique in algebraic geometry and optimization.", "---", "### Final Thoughts", "This problem beautifully illustrates how substitution of variables simplifies complex equations and reveals geometric relationships. Substituting $ z = x $ and $ y = a - x $ transforms a 3D surface equation into a manageable quadratic, allowing us to solve precisely for intersections.", "For students and practitioners, this serves as a valuable reminder: strategic substitution can uncover symmetry, simplify computation, and expose the true nature of geometric constraints.", "---", "Tagline for SEO:\nSolve quadratic surfaces with variable substitution — unlock geometry through algebra \nSphericalGeometry #AlgebraicSolutions #ConicSections #EquationSubstitution #3DGeometry", "---", "Keywords: $ x^2 + y^2 + z^2 = 2a^2 $, $ z = x $, $ y = a - x $, quadratic equations, geometric intersections, Algebra & Geometry, Trigonometric substitution, solve equations, parametric substitution.", "---", "Meta Description:\nLearn how substituting $ z = x $ and $ y = a - x $ into $ x^2 + y^2 + z^2 = 2a^2 $ reduces the equation to a solvable quadratic, revealing key geometric intersections. Perfect for algebra students and geometric problem Solvers.\n#Math #Geometry #EquationSolving"]