Solve $x = Cx+d$ for $x$

Question

Given

$$C = \begin{bmatrix} 0.7 & 0.2 \\ 0.1 & 0.5 \\ \end{bmatrix} \qquad \text{ and } \qquad d = \begin{bmatrix} 26\\ 52\\ \end{bmatrix}$$

Solve the equation $x = C x + d$ for $x$.

I'm a little confused as to why $x$ is on both sides of the equation. Am I supposed to find the determinant to solve this?

Update: I multiplied the identity matrix by matrix $C$. Then I multiplied this times $d$ to get $x$. Which gave me the answer:

\begin{bmatrix} 180\\ 140\\ \end{bmatrix}

Answer 1

Hint

Substitute $x=Ix=\begin{bmatrix}1&0\\0&1\end{bmatrix}x$ and solve $$(C-I)x+d=0$$

Answer 2

With $x=(u,v)$ the equation $x= Cx+d $ is equivalent to

$u=0.7u+0.2v+26$

$v=0.1u+0.5v+52.$

Can you take it from here?