1. ## Simultaneous Equations

I am having trouble with the following simultaneous equations

$x = \alpha x + \beta y$
$y = (1 - \alpha)x + (1 - \beta)y$
$x + y = 1$

I also have the solutions:

$x = \frac{\beta}{1 + \beta - \alpha}$ and $y = \frac{1 - \alpha}{1 + \beta - \alpha}$

but I don't know how these were worked out. Can someone please show me?

2. Originally Posted by MathTragic
I am having trouble with the following simultaneous equations

$x = \alpha x + \beta y$
$y = (1 - \alpha)x + (1 - \beta)y$
$x + y = 1$

I also have the solutions:

$x = \frac{\beta}{1 + \beta - \alpha}$ and $y = \frac{1 - \alpha}{1 + \beta - \alpha}$

but I don't know how these were worked out. Can someone please show me?
first let $x = 1 - y$

sub this into $y = (1 - \alpha)x + (1 - \beta)y$

$y = (1 - \alpha)(1-y) + (1- \beta) y$
$y = 1 - y - \alpha + \alpha y + y - y\beta$
$y + y - \alpha y - y + y \beta = 1 - \alpha$
$y (1 - \alpha + \beta) = 1 - \alpha$
$y = \frac{1 - \alpha}{1 + \beta - \alpha}$

and then sub this into the first equation and solve for x