We will use an example to answer this question. Let's consider a system of linear equations. {2x+y+8=122x+y+2=6(I)(II) Note that both the x- and the y-variables have the same coefficient in both equations. Thus, in order to use the elimination method, we will subtract Equation (II) from Equation (I). −2x+y+82x+y+26===121616 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. {x−y+2=6-x+y+3=5 In this example the x- and the y-variables have opposite coefficients. Therefore, to solve the system, we will add the equations. +x−y+2-x+y+35===6511 Since we know that 5=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.