1. Problem: Find the value of the determinant $$\begin{vmatrix} x + y & x & y \\ x & x + z & z \\ y & z & y + z \end{vmatrix}$$ 2. Formula and rules: The determinant of a 3x3 matrix $$\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)$$ 3. Calculate intermediate terms: - Calculate $ei - fh = (x+z)(y+z) - z \cdot z = (x+z)(y+z) - z^2$ - Calculate $di - fg = x(y+z) - z y = x(y+z) - zy$ - Calculate $dh - eg = x z - (x+z) y = xz - y(x+z)$ 4. Substitute into determinant formula: $$\det = (x+y)((x+z)(y+z) - z^2) - x(x(y+z) - zy) + y(xz - y(x+z))$$ 5. Expand terms: - $(x+z)(y+z) = xy + xz + yz + z^2$ - So, $(x+z)(y+z) - z^2 = xy + xz + yz$ - $x(y+z) - zy = xy + xz - zy$ - $xz - y(x+z) = xz - xy - yz$ 6. Substitute expansions: $$\det = (x+y)(xy + xz + yz) - x(xy + xz - zy) + y(xz - xy - yz)$$ 7. Expand all terms: - $(x+y)(xy + xz + yz) = x(xy + xz + yz) + y(xy + xz + yz) = x^2 y + x^2 z + xyz + xy^2 + xyz + y^2 z$ - $- x(xy + xz - zy) = -x^2 y - x^2 z + x z y$ - $+ y(xz - xy - yz) = y x z - y^2 x - y^2 z$ 8. Combine like terms: - $x^2 y$ and $-x^2 y$ cancel - $x^2 z$ and $-x^2 z$ cancel - $xyz + xyz + x z y + y x z = 4xyz$ - $xy^2$ remains - $y^2 z$ and $-y^2 z$ cancel - $- y^2 x$ remains 9. Final simplified form: $$\det = 4xyz + xy^2 - y^2 x = 4xyz + xy^2 - xy^2 = 4xyz$$ 10. Since $xy^2 - y^2 x = 0$, the determinant simplifies to: $$\boxed{4xyz}$$ Slug: determinant-xyz Subject: algebra desmos: {"latex":"y=4xyz","features":{"intercepts":true,"extrema":true}} q_count: 10