Answer
$$(0,0)=-\frac{2}{7}t(3,4)+\frac{8}{7}t(-1,1)+t(2,0) .$$
$$(2,0)=\frac{2}{7}(3,4)-\frac{8}{7}(-1,1) .$$
Work Step by Step
Consider the combination
$$a(3,4)+b(-1,1)+c(2,0) =0, \quad a,b,c \in R.$$
Which yields the following system of equations
\begin{align*}
3a-b+2c&=0\\
4a+b &=0.
\end{align*}
The augmented matrix matrix is given by
$$\left[\begin {array}{cccc} 3&-1&2\\ 4&1&0\end {array}
\right]$$
The reduced row echelon form can be calculated as follows
$$\left[\begin {array}{cccc}1&0&\frac{2}{7}\\ 0&1&-{\frac {8}{7}}
\end {array}
\right],$$
then we have the solution
$$a=-\frac{2}{7}t,\quad b=\frac{8}{7}t, \quad c=t.$$
Hence, we have the combinations
$$(0,0)=-\frac{2}{7}t(3,4)+\frac{8}{7}t(-1,1)+t(2,0) .$$
$$(2,0)=\frac{2}{7}(3,4)-\frac{8}{7}(-1,1) .$$
