Laplace transform for a non homogeneous system of eqns. does not work out

I tried the Laplace transform method advised in this post Inhomogenous Differential system , however it didn't work.

I have the following system:

$y'_1(t)=-y_1(t)-2y_2(t)+3$

$y_2'(t)=3y_1(t)+4y_2(t)+3$

and it seems rather simple accounting for the solution by Eerland in the original post. But something goes wrong. I do Laplace transforms of the first and second equations get:

$ℒ_t[y_1'(t)]=-(ℒ_t[x(t)](s)) - 2 (ℒ_t[y(t)](s)) + 3/s$

$ℒ_t[y_2'(t)]= (ℒ_t[x(t)](s)) + 4 (ℒ_t[y(t)](s)) + 3/s$

Then I thought I should eliminate one of the two, and therefore multiply the upper by 2, and sum them together, but this gives:

$2ℒ_t[y_1'(t)]+ℒ_t[y_2'(t)]=-(ℒ_t[x(t)](s))+\frac{15}{s}$

The initial conditions are

$y_1(0)=4$

$y_2(0)=5$

But from here I am stuck. How do I apply these initial conditions correctly to solve the last eqn?

Thanks

Your notation is confusing. I will use capital letters for the associated transforms.

$$sY_1(s)-y_1(0)=-Y_1(s)-2Y_2(s)+ \dfrac{3}{s}$$ $$sY_2(s)-y_2(0)=3Y_1(s)+4Y_2(s)+\dfrac{3}{s}$$

This is a linear system of two equations with two unknowns. Solve for $Y_1(s)$ and $Y_2(s)$ and calculate the inverse transform.