How to tell the number of solutions to a simultaeneous equation? 
How many solutions are there to the following simultaeneous equations?:
  $$
\begin{align}
x - 2y + 3z = 1\\
2x + 2y - z = 4\\
4x - y + 5z = 6
\end{align}
$$

How can I know the number of solutions that there are?
EDIT: 
I have found z = 0, y = 2/7, x = 11/7 as solutions but the answers I have say that there are infinite solutions. How can one deduce this?
Please note that I am a high school student and am unable to understand advanced mathematics.
 A: Find the augmented matrix corresponding to the system of equations, then, using Gaussian elimination, manipulate it into row echelon form. This will give you information about the number of solutions. 
A: Hint: multiplying the first equation by $-2$ and addin g to the second we get
$$6y-7z=2$$ (I)
multiplying the first by $-4$ and adding to the third we get
$$7y-7z=2$$ (II)
can yuo finish?
multiplying (I) by $-1$ and adding to (II):
we get $$y=0$$ and $$z=-\frac{2}{7}$$
i have got $$x=\frac{13}{7}$$ Make a Control if my solution is right
A: Write the system as $Ax=b$ with the matrix
$$
A=\begin{pmatrix} 1 & -2 & 3 \cr 2 & 2 & -1 \cr 4 & -1 & 5 \end{pmatrix}.
$$
Since $\det(A)\neq 0$ we can form the inverse, so that there is a unique solution $x=A^{-1}b$. Indeed, $x=A^{-1}(Ax)=A^{-1}b=(13/7,0,-2/7)^T$. In general, use row-echelon form, and see how many elements your field $K$ has (such equations do make sense over finite fields, too).
A: The general result  for a square matrix of dimension $n$ is the following:

  
*
  
*If the matrix has maximal rank $r=n$, there is exactly one solution.
  
*If the matrix has rank $r<n$, and if the augmented matrix has the same rank, the solution are an affine subspace of dimension $n-r$.
  
*If the matrix has rank $r<n$, and if the augmented matrix has rank $r'>r$, there is no solution.
  

