Answer
$$[x]_S=\left[ \begin {array}{ccc} 8\\ -2\end {array} \right].$$
Work Step by Step
Since we have
$$x= 3(1,1)+5(1,-1)=(8,-2).$$
then we can write $x$ relative to the standard basis of $R^2$ as follows
$$x= (8,-2)=8(1,0)-2(0,1) .$$
Thus, he coordinates matrix of $x$ in $R^2$ relative to the
standard basis is
$$[x]_S=\left[ \begin {array}{ccc} 8\\ -2\end {array} \right].$$
