Note: This refers to the system as original given.
Rewrite the equations: $\begin{array}{rcccccl} &&3y&-&4z&=&0\\ 2x&-&4y&+&3z&=&0\\ 2x&+&y&-&3z&=&0 \\ x&+&y&+&z&=&1 \end{array}$ Now simply solve the system: $\begin{align*} \left(\begin{array}{rrr|c} 0 & 3 & -4 & 0\\ 2 & -4 & 3 & 0\\ 2 & 1 & -3 & 0\\ 1 & 1 & 1 & 1 \end{array}\right) &\to \left(\begin{array}{rrr|r} 1 & 1& 1 & 1\\ 0 & 3 & -4 & 0\\ 2 & -4 & 3 & 0\\ 2 & 1 & -3 & 0 \end{array}\right) \to\left(\begin{array}{rrr|r} 1 & 1 & 1 & 1\\ 0 & 3 & -4 & 0\\ 0 & -6 & 1 & -2\\ 0 & -1 & -5 & -2 \end{array}\right)\\ &\to \left(\begin{array}{rrr|r} 1 & 1 & 1 & 1\\ 0 & 1 & 5 & 2\\ 0 & 3 & -4 & 0\\ 0 & -6 & 1 & -2 \end{array}\right) \to \left(\begin{array}{rrr|r} 1 & 1 & 1 & 1\\ 0 & 1& 5 & 2\\ 0 & 0 & -19 & -6\\ 0 & 0 & 31 & 10 \end{array}\right). \end{align*}$ The last two lines tell us that the system is inconsistent, so the system has no solution.
Alternatively: plugging in $\frac{4}{3}z$ for $y$, we obtain from the second equation $\begin{align*} 2x - \frac{16}{3}z + 3z &= 0\\ 2x -\frac{7}{3}z &= 0\\ 6x -7z &=0.\end{align*}$ The third equation gives $\begin{align*} 2x +\frac{4}{3}z - 3z &= 0\\ 2x -\frac{5}{3}z &=0\\ 6x - 5z&=0 \end{align*}$ Since $6x=7z$ and $6x=5z$, then $x=z=0$, hence $y=0$, which makes $x+y+z=1$ impossible.
If the system is meant to be
$\begin{array}{rcccccl} -4x&+&3y&&&=&0\\ 2x&-&4y&+&3z&=&0\\ 2x&+&y&-&3z&=&0 \\ x&+&y&+&z&=&1 \end{array}$ then proceed as above to solve the system.
$\begin{align*} \left(\begin{array}{rrr|r} -4 & 3 & 0 & 0\\ 2 & -4 & 3 & 0\\ 2 & 1 & -3 & 0\\ 1 & 1 & 1 & 1 \end{array}\right) &\to \left(\begin{array}{rrr|r} 1 & 1 &1 & 1\\ -4 & 3 & 0 & 0\\ 2 & -4 & 3 & 0\\ 2 & 1 & -3 & 0 \end{array}\right) \to \left(\begin{array}{rrr|r} 1 & 1 & 1 & 1\\ 0 & -5 & 6 & 0\\ 2 & -4 & 3 & 0\\ 0 & 5 & -6 & 0 \end{array}\right)\\ &\to \left(\begin{array}{rrr|r} 1 & 1 & 1 & 1\\ 0 & -5 & 6 & 0\\ 0 & -6 & 1 & -2 \end{array}\right) \to \left(\begin{array}{rrr|r} 1 & 1 & 1 & 1\\ 0 & 1 & 5 & 2\\ 0 & -5 & 6 & 0 \end{array}\right)\\ &\to \left(\begin{array}{rrr|r} 1 & 1 & 1 & 1\\ 0 & 1 & 5 & 2\\ 0 & 0 & 31 & 10 \end{array}\right) \to \left(\begin{array}{rrr|r} 1 & 0 & -4 & -1\\ 0 & 1 & 5 & 2\\ 0 & 0 & 1 & \frac{10}{31} \end{array}\right)\\ &= \left(\begin{array}{rrr|r} 1 & 0 & 0 & \frac{9}{31}\\ 0 & 1 & 0 & \frac{12}{31}\\ 0 & 0 & 1 & \frac{10}{31} \end{array}\right). \end{align*}$ So the solution is
$x = \frac{9}{31}$,
$y=\frac{12}{31}$,
$z=\frac{10}{31}$.