When do two simultaneous equations have a solution?

This question was put to someone during their teacher training. This is a good question to ask, it is very important in maths to know that something you intend to study does actually exist. To explain the question we look at an example.Suppose that we have simultaneous equations x+2y=0 (1) 3x+4y=0 (2) A solution to these equations means values of x and y that when substituted into (1) and (2) satisfy the equations.Notice that if x=0 and y=0 then the equations are satisfied. The question is are there any other solutions, any other values of x and y that satisfy the equations?The answer is no in this case, and there is a simple method to check. If we have two equations like (1) and (2), we multiply the numbers in front of x and y in opposing equations and subtract the result. For (1) and (2) we would have 14=4 and 23=6 Then the number we want is 4-6=-2. This is the number that is important. It is not zero, so there are no solutions other than x=0, y=0 to the equations. Another example: Are there solutions other than x=0, y=0 to the equations below? 4x+7y=0 (3) 5x+9y=0 (4) The number we need to find in this case is then 49-57=36-35=1. This is not zero, so there are no other solutions. This method applies to all equations that look like ax+by=0 cx+dy=0 where a, b, c, and d are just numbers. If ad-bc is not zero, then there are no solutions to these equations other than x=0, y=0.It is very important to remember that if we had equations like 3x+4y=2 and 5x+6y=5 we could not apply this method, because x=0, y=0 is not even a solution in this case.