# Solving 4 unknowns with 4 equations all equal to zero

I have the following equations:

\begin{align} a&=0.35a+0.35d\\ b&=0.65a+0.65d\\ c&=0.35b+0.35c\\ d&=0.65b+0.65c \end{align}

I know $b=d$, but where do I go from here?

I have a similar equation that I know the answer to, and I need to solve this one in the same way. The other question is:

\begin{align} a&=0.8a+0.5c\\ b&=0.2a+0.5c\\ c&=0.5b+0.2d\\ d&=0.5b+0.8d \end{align}

The answers to these equations are $a=d=\dfrac5{14}$ and $b=c=\dfrac2{14}$.

I don't know how to get there though.

• How have you found $b=d$ and what else have you tried? Commented Mar 6, 2015 at 21:17
• The two systems of equations you wrote are indetermined, i.e. they have not a unique solution. For example, you have the trivial solution $a=b=c=d=0$, which is not the solution you posted. Commented Mar 7, 2015 at 0:15
• Worked it out now, they are probabilities so a+b+c+d=1 Sorry! Commented Mar 7, 2015 at 17:17

From first equation: $0.65 a = 0.35 d$ (X), from third: $0.65c = 0.35 b$. Therefore (combining (X) with second equation) we have $b = d$. So the solution is
$$(a,b,c,d) = \left(\frac{7t}{13}, t,\frac{7t}{13},t \right), \quad t \in \mathbb R$$