Does existence of independent variables ensure that there are infinitely many solutions? As I stated above. In a system of equations, does existence of independent variables ensure that there are infinitely many solutions?
THANK YOU!

I assume you mean independent variable as to mean the same as my $t$ in the following example: 
Say we a system of equations
\begin{align}
-x_1-2x_2-5x_3&=-3 \\
2x_1+3x_2+8x_3&=4 \\
2x_1+6x_2+14x_3&=10 \\
\end{align}
which we write on matrix form and then reduce:
\begin{align}
\pmatrix{
-1 & -2 & -5 & -3 \\
2 & 3 & 8 & 4 \\
2 & 6 & 14 & 10
} \rightarrow 
\pmatrix{
1 & 2 & 5 & 3 \\
1 & 1 & 2 & 2 \\
0 & 0 & 0 & 0
}
\end{align}
which is the same as 
\begin{align}
x_1+2x_2+5x_3&=3 \\
x_2+2x_3&=2
\end{align}
Since we have two equations with three variables, we have a free parameter, which we define as $t=x_3 \in \mathbb{R}$.
Parameterization gives us that the full solution can be written as
$$\pmatrix{x_1 \\ x_2 \\ x_3}=\pmatrix{-1 \\2 \\ 0} +t\pmatrix{-1 \\ -2 \\ 1}$$
which indeed represents an infinite number of points and therefore solutions. 
