I have the following system of linear equations with 5 variables, but I do not know how to find the solutions because there are only 3 equations. I have thought about using reduced row echelon forms, but it seems like there are no solutions?

$$−x_1 + x_3 − x_4 + x_5 = 1\\ x_1 + x_2 + 2x_3 − 3x_5 = 3\\ 2x_1 + x_2 + x_3 + x_4 − 4x_5 = 2$$

to reduced row echelon form:

-1 0 1 -1 1 1

0 1 3 -1 -2 4

0 1 3 -1 -2 4

  • pick any $x_1$ and $x_2$ and then solve it as system of three unknowns with three equations. You will have infinite number of solutions. – Vasya Oct 11 at 12:45
  • I get that there are infinite solutions, but how do you find the general solution? – Yolanda Hui Oct 11 at 13:41
  • treat $x_1$ and $x_2$ as constants and solve the system. You will have $x_3, x_4, x_5$ expressed in terms of $x_1$ and $x_2$. That will be the general solution. – Vasya Oct 11 at 13:50
  • why do we treat x1 and x2 as constants? can't we choose x3, x4 or x5 instead? – Yolanda Hui Oct 11 at 13:51
  • 1
    Yes, basically you pick any values for $x_3, x_4, x_5$ and calculate $x_1, x_2$ to get one of the solutions. – Vasya Oct 11 at 14:46

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