The graph of a linear equation with three variables, typically written in the form $$ax + by + cz = d$$, represents a flat, two-dimensional plane embedded within a three-dimensional Cartesian coordinate space. Every point on this plane corresponds to a unique ordered triple $$(x, y, z)$$ that acts as a valid solution directly satisfying the equation.

Claude

A linear equation involving three variables is an algebraic expression that can be written in the standard form $$ax + by + cz = d$$. In this structure, the coefficients $$a$$, $$b$$, $$c$$, and the constant $$d$$ must be real numbers, with the additional restriction that $$a$$, $$b$$, and $$c$$ cannot all simultaneously be equal to $$0$$.

Linear Equation in Three Variables

https://openstax.org/details/books/intermediate-algebra-2e

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