# Approximation to Solution of a Linear System of Equations

Consider a linear system, $Ax=b$ where $A$ is the coefficient matrix, $x$ is the unknown vector of variables, and $b$ is a vector of constants in which all entries are same.

Is there a way to find out relative values of $x$ without actually solving the system of equations? In other words, can we say by just looking at $A$ that $x_1$ will be greater than $x_2$ because the sum of entries of first row is greater than the sum of entries of the 2nd row of $A$.

I have intentionally made the assumption that all entries of $b$ are same, so that they have no effect on our approximate solution.

• If you have $A x = b$ and you would like to check if the solution satisfies, say, $x_1 \geq x_2 \geq \cdots \geq x_n$, then you can use linear programming. – Rodrigo de Azevedo Mar 12 '18 at 12:29
• Why don't you want to solve the system? – rviertel Mar 12 '18 at 19:40

A counterexample of what you propose: $$A = \left[\begin{matrix} 1 & 1&0\\0 & 1& 0\\ 0 & 0 &1\end{matrix}\right], \quad b=\left[\begin{matrix}1\\1\\1\end{matrix}\right]$$
The sum of he first row of $A$ is greater than the rest and $x_1=0<x_2=x_3=1$