The given equation is: $x\begin{bmatrix}1\\ 2\end{bmatrix}+y\begin{bmatrix}2\\ 5\end{bmatrix}=\begin{bmatrix}4\\ 9\end{bmatrix}$

Performing scalar multiplication, we get: $\begin{bmatrix}x\\ 2x\end{bmatrix}+\begin{bmatrix}2y\\ 5y\end{bmatrix}=\begin{bmatrix}4\\ 9\end{bmatrix}$

Adding the matrices, we have: $\begin{bmatrix}x+2y\\ 2x+5y\end{bmatrix}=\begin{bmatrix}4\\ 9\end{bmatrix}$

Equating the corresponding elements, we get the following system of linear equations:

1) $x + 2y = 4$

2) $2x + 5y = 9$

We can solve this system of equations using substitution or elimination. Let's use elimination.

Multiply equation (1) by 2: $2x + 4y = 8$

Subtract this new equation from equation (2): $(2x + 5y) - (2x + 4y) = 9 - 8$

This simplifies to: $y = 1$

Substitute $y = 1$ into equation (1): $x + 2(1) = 4$

So, $x + 2 = 4$, which gives $x = 2$