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## Homework Statement

In the following problem, find conditions on a and b such that the system has no solution, one solution, and infinitely many solutions.

x + by = -1

ax + 2y = 5

## Homework Equations

None that I know of.

## The Attempt at a Solution

Basically, first I put the entire equation into a matrix.

[ 1 b | -1

a 2 | 5 ]

I reduce the bottom by subtracting R2 - aR1

[ 1 b | -1

0 2-ab| 5 +a ]

I then reduce the bottom again by dividing R2/(2-ab)

[ 1 b | -1

0 1 | (5+a)/(2-ab)]

I remove the b from the top by subtraction: R1 - bR2

[1 0 | -1 - b((5+a)/(2-ab))

0 1 | (5+a)/(2-ab) ]

This leaves me with the values for x and y, and for the first question I am correct in saying if ab = 2, then there is no solution as it is undefined. However, my unique solution is somehow wrong and I would like some help in determining if I made an error or I somehow didn't reduce something.

The correct unique solution is: x = (-2 - 5b)/(2-ab) y = (a+5)/(2-ab)

Also, I have no idea what finding an infinite solution means, I would really like some help on clarifying that. Thank you.