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## Homework Statement

So the actual problem "Find the value of a for which the following system of linear equations has a solution"

2x + 4y + z = a

-4x -7y + 0 = 1

0 -1y -2z = 1

## Homework Equations

## The Attempt at a Solution

I thought one approach was to find a basis for the corresponding matrix and see what value of a would make that vector in the space formed by the basis. That is, see what value of a would make (a, 1, 1) in the range of the matrix.

But when I row reduce I get

2 4 1

0 1 2

0 0 0

So I want to say the 1st and 2cnd columns form a basis -> (2, -4, 0) and (4, -7, -1). But then some linear combination of these should equal (1, 0, -2). However, the first components of the basis appear to never be able to combine linearly to 1. That is there are no integers x,y such that 2x + 4y = 1 since -> x + 2y = 1/2.....? I feel like I must be making a really trivial mistake?!?!