Solution

Elimination is a fundamental step in solving systems of equations, but the method demonstrated on that page relies upon the right hand side being known values. This page instead supposes that the right hand side is unknown.

Introduction

w + 2x + 2y + 2z = a
2w + 4x + 6y + 8z = b
3w + 6x + 8y + 10z = c

This corresponds to the following eliminated augmented matrix:

┌                  ┐
│ [1] 2  2  2     a│
│  0  0 [2] 4  b-2a│
│  0  0  0  0 c-b-a│
└                  ┘

The immediate question then is: for what values of a, b, and c is this system solvable?

Trivial Solvability

The above system can be easily shown to be solvable for some values.

0w + 0x + 0y + 0z = c - b - a
0 = c - b - a

In other words, for any values of a and b which sum to c, this system is solvable.

General Solvability

Solutions

The first step to computing the complete solution (x_c) is computing a particular solution (x_p).

Find a valid set of values for the right hand side. In this system, for example, [1 5 6].

Set all free varaibles to zero. Substitute to find all pivot variables.

2y = 3
y = 3/2

w + 2y = 1
w + 2(3/2) = 1
w + 3 = 1
w = -2

The particular solution is:

[-2 0 3/2 0]

Next, compute the null space. If the matrix is full rank, the particular solution is the complete solution.

Finally, the complete solution is the particular solution plus any combination of the null space.

┌    ┐     ┌    ┐     ┌    ┐
│ -2 │     │ -2 │     │  2 │
│  0 │ + C │  1 │ + C │  0 │
│ 3/2│    1│  0 │    2│ -1 │
│  0 │     │  0 │     │  1 │
└    ┘     └    ┘     └    ┘

The values C₁ and C₂ are any number, because any combination of the two vectors is valid for the complete solution.

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