1. ## Simultaneous Equations

I am having trouble with the following simultaneous equations

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

I also have the solutions:

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

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

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

I also have the solutions:

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

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

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

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

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