Describing Solutions to Systems of Linear Equations

We will use an example to answer this question. Let's consider a system of linear equations. $$\{\begin{array}{ll}2x+y+8=12& \text{(I)}\\ 2x+y+2=6& \text{(II)}\end{array}$$ Note that both the $x\text{-}$ and the $y\text{-}$variables have the same coefficient in both equations. Thus, in order to use the elimination method, we will subtract Equation (II) from Equation (I). $$\begin{array}{crcl}& 2x+y+8& =& 12\\ -& 2x+y+2& =& 6\\ & 6& =& 6\end{array}$$ We arrived to a true statement. Its truthfulness does not depend on any variable. Thus, any value for $x$ and $y$ will satisfy the statement. This happens when the two lines are coincidental and then the system has infinitely many solutions.

Finally, to answer this question, we will use a third example. $$\{\begin{array}{l}x-y+2=6\\ \text{-}x+y+3=5\end{array}$$ In this example the $x\text{-}$ and the $y\text{-}$variables have opposite coefficients. Therefore, to solve the system, we will add the equations. $$\begin{array}{crcl}& x-y+2& =& 6\\ +& \text{-}x+y+3& =& 5\\ & 5& \ne & 11\end{array}$$ Since we know that $5\ne 11,$ we have a arrived at a false statement. There are no values for $x$ and $y$ that satisfy it. Thus, the system has no solution. This means the lines do not intersect. Every time we arrive to a false statement, the system has no solution.