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I was studying determinants, and the topic that came up was the 'eliminant' of a system of equations. My book says that the eliminant of the system of equations:

$$a_1x + b_1y + c_1z = 0$$

$$a_2x + b_2y + c_2z = 0$$

$$a_3x + b_3y + c_3z = 0$$

is

$$\begin{vmatrix}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3\end{vmatrix}=0$$

Here, three equations are required to eliminate three variables from the system of equations. However, if we know $z=1$, we get three equations in two variables. But we still need at least three equations to be able to eliminate $x, y$.

I don't understand the logic behind this. Precisely how many equations are needed to eliminate all the variables? Is there any intuitive way to understand the process of elimination (not how to eliminate but what elimination actually means) and see the reason for this?

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