1. **State the problem:** We are given the equation of a quadratic surface: $$x^2 + y^2 + z^2 - xy - yz - zx = 0$$ and two linear expressions: $$5x - 4y + 8z$$ and $$3y$$ We need to analyze or solve this system or understand its geometric meaning. 2. **Simplify the quadratic expression:** Rewrite the quadratic expression in a more symmetric form. Note that $$x^2 + y^2 + z^2 - xy - yz - zx = \frac{1}{2} \left((x - y)^2 + (y - z)^2 + (z - x)^2\right)$$ This is a classic identity expanding the squared terms: $$ (x - y)^2 = x^2 - 2xy + y^2 $$ $$ (y - z)^2 = y^2 - 2yz + z^2 $$ $$ (z - x)^2 = z^2 - 2zx + x^2 $$ Adding all: $$ (x - y)^2 + (y - z)^2 + (z - x)^2 = 2(x^2 + y^2 + z^2 - xy - yz - zx) $$ Hence, $$x^2 + y^2 + z^2 - xy - yz - zx = 0 \iff (x - y)^2 + (y - z)^2 + (z - x)^2 = 0$$ 3. **Interpret this condition:** The sum of squares of real numbers is zero if and only if each term is zero: $$x - y = 0$$ $$y - z = 0$$ $$z - x = 0$$ Which means $$x = y = z$$ 4. **Conclusion on values:** The quadratic equation constrains the point $(x, y, z)$ to lie on the line $$x = y = z$$ 5. **Evaluations of the linear expressions on the line:** Substitute $y = x$, $z = x$: $$5x - 4y + 8z = 5x - 4x + 8x = (5 - 4 + 8)x = 9x$$ $$3y = 3x$$ 6. **Summary:** - The original quadratic surface reduces to the line $x=y=z$. - The two linear expressions become linear functions of $x$ along this line. **Final answer:** On the locus defined by the quadratic equation (the line $x=y=z$), $$5x - 4y + 8z = 9x$$ and $$3y = 3x$$.