# Solving for a Specific Variable in an Underdetermined System of Linear Equations

I have an underdetermined system of linear equations of the form $$Ax = b$$, where:

$$A = \begin{bmatrix} a_{20} & a_{10} & a_{00} & 0 & 0 \\ 0 & a_{21} & a_{11} & a_{01} & 0 \\ 0 & 0 & a_{22} & a_{12} & a_{02} \\ \end{bmatrix}, \quad x = \begin{bmatrix} x_{0} \\ x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ \end{bmatrix},\quad b = \begin{bmatrix} b_{0} \\ b_{1} \\ b_{2} \\ \end{bmatrix}$$

and

I am interested in solving for $$x_2$$. Is it possible to get a unique solution for $$x_2$$ in this underdetermined system?

Additionally, suppose $$x_0$$ and $$x_1$$ are known to us. How would this affect the solution for $$x_2$$? I am interested in both the general solution where $$x_0$$ and $$x_1$$ are not known, and the specific solution where $$x_0$$ and $$x_1$$ are known.

Any help or guidance would be greatly appreciated!

Since $$A$$ is not full column rank, it has nontrivial nullspace, i.e., $$\mathcal{N}(A) \neq \{0\}$$. Hence, any vector in the affine set $$\{x \ |\ x = \bar{x} + w, \ A\bar{x} = b,\ \bar{x}\in \mathcal{R}(A^T),\ \forall w\in\mathcal{N}(A)\}$$ is a solution to your system. Here, $$\bar{x}$$ is the unique solution to the system in the domain of $$A$$, $$\mathcal{R}(A^T)$$.
If $$a_{00}\neq 0$$ and you know $$(x_0, x_1)$$, then from the first equation $$a_{20}x_0 + a_{10}x_1+ a_{00}x_2 = b_0$$, you can uniquely determine $$x_2$$.