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]

The augmented matrix of the system of equations is

which row-reduces to

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}, so rearranging the equations represented by the two nonzero rows to gain expressions for the variables x₁ and x₂ yields the solution set,