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## Homework Statement

1. Express the general solution of the following system in vector form.

2x + 4y + 6z + 4w = 4

2x + 5y + 7z + 6w = 3

2x + 3y + 5z + 2w = 5

Clearly identify the particular solution. Also produce a vector with no zero components that satisfies the corresponding homogeneous system.

## Homework Equations

## The Attempt at a Solution

The solved echelon form is:

2x + 4y + 6z + 4w = 4

0x + 1y + 1z + 2w = -1

x = 4 + 2w -z

y = -1 - z - 2w

solution set: (4-z+2w, -1-z-2w, z, w)

The solution in vector form is displayed as:

x |0 | |4 | | -1 | | 2|

y = |0 |+ |-1|+ | -1 |z + | 2|w

z |0 | |0 | | 1 | | 0|

w |0 | |0 | | 0 | | 1|

The particular solution is the vector:

|4 |

|-1|

|0 |

|0 |

What I'm confused by is this statement: "Also produce a vector with no zero components that satisfies the corresponding homogeneous system.". What is the corresponding homogeneous system? How do I find it? How do I then produce this vector with no zero components?

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