# System of linear equations with alpha variable

1. May 27, 2010

### literacola

1. The problem statement, all variables and given/known data

For which value of alpha does the following system of linear equations have a solution?

x + y = 2
x +αy = -1
αx + y = 1

3. The attempt at a solution

I put it into a matrix that looks like this:

1 1 | 2
1 α | -1
α 1 | 1

Then I subtracted row1 from row2 and it made the matrix look like:

1 1 | 2
0 (α-1) | -1
α 1 | 1

Then I took the 2nd row as α -1 = -1 and solved α to be zero. So, does this mean that the solution for this is all numbers α such that α is not equal to zero?

2. May 27, 2010

### gabbagabbahey

First, when you subtract the first row from the second you get

$$\left[\begin{array}{cc|c} 1 & 1 & 2 \\ 0 & \alpha-1 & -3 \\ \alpha & 1 & 1 \end{array}\right]$$

Second, your second row tell you $(\alpha-1)y=-3$, not $(\alpha-1)=-3$

No.

3. May 27, 2010

### literacola

Cool, thanks! So what's the next step? Not sure where to go from here...

4. May 27, 2010

### literacola

Can I get a hint?

5. May 27, 2010

### gabbagabbahey

I think the easiest method is to simply say that if $(\alpha-1)y=-3$, then $y=\frac{3}{1-\alpha}$ and substitute that into any of your 3 equations, solve for $x$ and sub those into the remaining equations to get an equation for $\alpha$.