Answer
$$[x]_{B'}=\left[ \begin {array}{ccc} 2\\ -4\\3 \end {array} \right].$$
Work Step by Step
Writing $x$ as a linear combination of the basis $B'$ as follows
$$x=\left(3,-\frac{1}{2},8\right)= a\left(\frac{3}{2},4,1\right)+b\left(\frac{3}{4},\frac{5}{2},0\right)+c\left(1,\frac{1}{2},2\right).$$
We get the system
\begin{align*}
8a+7b+c&=3\\
11a+4c&=19\\
10b++6c&=2.
\end{align*}
By solving the above system we have the soluiton
$$a=2, \quad b=-4, \quad c=3.$$
Thus, the coordinate matrix of $x$ in $R^3$ relative to the
basis $B'$ is
$$[x]_{B'}=\left[ \begin {array}{ccc} 2\\ -4\\3 \end {array} \right].$$
