Why does the reduced matrix $\left(\begin{array}{ccc|c}0&1&0&-7\\ 0&0&1&10\end{array}\right)$ have infinitely many solutions? $$\left(\begin{array}{ccc|c}0&1&0&-7\\ 0&0&1&10\end{array}\right)$$
I thought the requirement for a matrix to have a unique solution was that when every variable is leading. It seems like both 1's in the above matrix are leading and the other variables are 0. So why doesn't this matrix have only a unique solution?

This is equivalent to the system:
$$\begin{eqnarray*}
  {0x + y + 0z}&=&{-7} \\ 
  {0x + 0y+ z}&=&{10} 
\end{eqnarray*}$$
Whatever $x$ you choose will work because $0x=0$ for all $x$. If the system were:
$$\begin{eqnarray*}
  { y + 0z}&=&{-7} \\ 
  { 0y+ z}&=&{10} 
\end{eqnarray*}$$
Then there is only one solution. It is very common for people to forget the $0x_i$ matter, a lot of people answer a different problem because they write the first system as the second system. The first system has infinite solutions, the second system as one solution. 
I suggest you to work on the equivalence between matrices and systems of linear equations and see what each result in matrices means in terms of results of systems of linear equations and vice-versa.
