How do you determine whether a linear system has one solution, many solutions, or no solution when given 3x-8y=0 and -2x+5y=-2?
1 Answer
I found
Explanation:
A fast way to "see" the various possibilities of solutions of your system is to look at the numerical coefficients remembering that, basically, here you are dealing with two equations representing two straight lines (that can cross each other, be parallel or one on top of the other):
Case 1: the coefficients are all the same:
example:
In this case you have the same equations representing the same lines.
Although a bit weird you can see that a line crosses itself in an
Case 1a: the coefficients of one equation are multiples of the ones of the other (i.e., they are multiplied by a number):
example:
The coefficients of the second equations are equal to the ones of the first multiplied by
Case 2: the coefficients of
example:
In this case the two lines are PARALLEL !!! So, they cannot cross and so they will not have solutions in common and you will have a system with no solutions.
Case 3: The coefficients of
example:
In this case the two lines will cross in one point of coordinates
isolate
substitute into the second:
back into: