Is this the correct solution?

Is this the correct solution?

Determine the coordinates of the vector $U=(4,5,-3)\;\text{of}\; R^3$ with
respect to base ${(1,0,0), (0,1,0), (0,0, 1)}$
$$x(1,0,0) + y (0,1,0) + z (0,0,1) = (4,5, -3)$$ $$(x, 0,0) + (0, y, 0) +
(0,0,z) = (4,5, -3)$$ $$x +0 +0 = 4 \Longrightarrow x = 4$$ $$0 +0 + y = 5
\Longrightarrow y = 5$$ $$0 +0 + z = -3 \Longrightarrow z = -3$$
Thus, the coordinates are $$(4,5, -3).$$ $$$$ $$$$And the base
$\{(1,1,1),(1,2,0),(3,1,0)\}$, just find the values ​​of $x$,
$y$ and $z$ such that $$x(1,1,1)+y(1,2,0)+z(3,1,0)=(4,5,-3)??$$
$$(x,x,x)+(y,2y,0)+(3z,z,0)=(4,5,-3)$$$\begin{cases}x+y+3z=4&\\
x+2y+z=5&\\
x+0+0=-3&\end{cases}$$$x=-3$$$$y=\frac{53}{5}$$and$$z=-\frac{6}{5}$$ Is
this correct?