# Matrix with 2 unknowns

• Temp0
In summary, Temp0 found a solution to the homework equation that was different from the one provided in the textbook. He is unsure of why this is the case and needs help from the forum to figure out why.

## 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.

Hi Temp0! Welcome to PF! Temp0 said:
[1 0 | -1 - b((5+a)/(2-ab))
0 1 | (5+a)/(2-ab) ]

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

That is the same as yours! Also, I have no idea what finding an infinite solution means

Hint: if A and B are two solutions, what can you say about A - B ? Also, I have no idea what finding an infinite solution means, I would really like some help on clarifying that. Thank you.
For example: if you are given graphs of f(x) and g(x) [neither contain discontinuities] and g(x) 's graph never "falls under" or crosses f(x) , no matter what the argument, g(x) has always higher values than f(x) and you are asked to provide solutions for g(x) > f(x) then you can say that there are infinitely many solutions. But you are dealing with an equal sign so that must mean the 2 graphs are...?

tiny-tim said:
Hi Temp0! Welcome to PF! That is the same as yours! Hint: if A and B are two solutions, what can you say about A - B ? Hmm, thanks for the help on the infinite thing, I finally get that ^^. However, I've tried reducing and expanding my answer, but it never becomes the same as the answer in the book. Are there any hints you can give me? =D

edit: nvm, I just looked at it again and realized how to get to the answer, thanks for your help =p.