Somebody hinted that one can solve a system of equations with some method of rotation. This is only needed for numeric solutions, otherwise Gauss-Jordan (RREF) is just fine.
What is this rotation about?
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It's called a Givens rotation. Similarly to Gaussian elimination, you can use a Givens rotation to eliminate a variable from a system of linear equations.
The basic idea is to find a 2D rotation matrix that will rotate a coefficient in the linear system to lie on the x-axis, thus eliminating it.
Note that this method is not trivially immune to numerical problems. If you directly compute the rotation angle, then you may have continuity issues. If you instead compute cos(alpha) and sin(alpha), you may risk division by zero if the radius is too small. Here is a numerically stable way to compute the rotation.
Givens rotations are also used for computing a QR decomposition: the triangular R is created by a series of eliminations by Givens rotations. Q is simply the product of all the rotation matrices, which are orthogonal.