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Suppose we have three simultaneously equations with $4$ variables with coefficients in finite fields, i.e.

$$\alpha_1A_1 + \beta_1B_1+\gamma_1C_1 + \theta_1D_1=x$$ $$\alpha_2A_1 + \beta_2B_1+\gamma_2C_1 +\theta_2D_1=y$$ $$\alpha_3A_1 + \beta_3B_1+\gamma_3C_1 +\theta_3D_1=z$$

Is it possible to solve for $A_1,B_1,C_1,D_1$ solely from equations above? Maybe we can use some properties of finite fields?

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    $\begingroup$ You can solve such a system in the same way you solve systems of linear equations with real or complex coefficients, for example by transforming it to echelon form, which allows you to see whether there are solutions at all. Note that the algorithm to pass to the echelon form only reuires operations that are possible in every field. $\endgroup$ – Hagen Knaf Jul 10 '14 at 10:07
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When dealing with linear equations, the field does not make much of a difference. All the methods and results you usually use are valid regardless of the field.

Specifically, you can never get a unique solution to $3$ equations in $4$ variables, even if the field is finite.

As already mentioned in the comments, you can reduce to echelon form to see if it has solutions, just like you would do over any other field.

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