*Unique solution:*
x + by = -1 .... (1)

ax + 2y = 5 .... (2)

Solve using the elimination method:

Multiply equation (1) by a, subtract it from equation (2) and then solve for y:

.

Note that this solution is OK

*provided the denominator is NOT equal to zero*, that is,

.

Now sub

into either equation and solve for x. Using equation (1):

.

So there's a unique solution, given by the above, when

.

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No unique solution: .

When there's no unique solution, that means that either there is an infinite number of solutions or no solution at all. To test which of these happens when ab = 2, you might do something like the following:

. Substitute this into you equations and do some re-arranging:

Equation (1):

.

Equation (2):

.

Now compare equations (1) and (2). You can see that when

the equations are the same. One equation and two unknowns => infinite number of solutions.

So when ab = 2 AND a = -5, there an infinite number of solutions.

It follows that there will be no solution at all when ab = 2 AND

.

When a = -5 and ab = 2,

. So the equations boil down to -5x + 2y = 5. The form of the infinite number of solutions can be expressed in several ways. To see where the books answer came from, note that if you let y = t, where t can be any real number:

.