find all solutions to the system of linear equations. Use your favorite computing device to row-reduce the augmented matrices for the systems, and write the solutions as a set, using correct set notation

Contributed by Robert Beezer Statement [43]

We row-reduce the augmented matrix of the system of equations,

In the spirit of Example SAA, we can express the infinitely many solutions of this system compactly with set notation. The key is to express certain variables in terms of others. More specifically, each pivot column number is the index of a variable that can be written in terms of the variables whose indices are non-pivot columns. Or saying the same thing: for each i in D, we can find an expression for x_i in terms of the variables without their index in D. Here D = {1, 2, 4}, so

x₁ + 2x₃ + 5x₅ = 6  →  x₁ = 6 − 2x₃ − 5x₅

x₂ − 3x₃ − 2x₅ = 9  → x₂ = 9 + 3x₃ + 2x₅

x₄ + x₅ = −8 → x₄ = −8 − x₅

As a set, we write the solutions precisely as