3a. You can tell if a system of linear equations has no solutions if the lines are parallel (do not intersect). They will have infinite solutions if the lines graph as the same line (intersect at every point).
3b. You can tell the system of linear equations has no solution if the equations result in a contradiction, such as 0 = 1. They will have infinite solutions if the result is a tautology, such as x = x or 0 = 0.
Word Problem
(a) There is actually no problem stated. If we assume the problem is to determine the length and width that meet the described conditions, then it would be appropriate to let variables represent the length (L) and width (W). The system of equations restates what the words state:
.. L - W = 2 . . . . . . length and width differ by 2 units
.. 2(L + W) = 40 . . the perimeter is 40 units
(b) You can solve this many ways. One way is to divide the second equation by 2, so you have a "sum and difference" problem
.. L - W = 2
.. L + W = 20
This is easily solved by adding the two equations to get
.. 2L = 22
.. L = 11
.. W = L -2 = 9
The solution is (L, W) = (11, 9). This is interpreted to mean the length of the rectangle is 11 units; the width is 9 units.
