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Question
Find conditions on a and b such that the following system of linear equations has
(a) no solutions,   (b) exactly one solution, and   (c) an infinite number of solutions

x+2y=3
ax+by= -9

Answer
Questioner:Vanessa
Country:Quebec, Canada
Category:Advanced Math
Private:No
Subject:Linear Algebra
Question:

Find conditions on a and b such that the following system of linear equations has
(a) no solutions,   (b) exactly one solution, and   (c) an infinite number of solutions

x+2y=3
ax+by= -9
.................................
If you attempt to solve using the standard techniques (eliminate x from the equations) you will get:

   3a + 9
y = --------
   2a - b

(and use that to get an expression for x)


Now if  2a - b /= 0 all is well and you will have one solution.

But if  2a - b = 0 all is not so well

This happens if  b = 2a

if  3a + 9 /= 0, then there is no solution at all.

if  3a + 9 = 0  you will have many solutions.
You can take it from there.

...............................

Suggestion: You have studied Cramer's rule?

D = the determinant:

|  1   2  |
|  a   b  |

= b - 2a

And the solution for x will be:

|  3   2  |
| -9   b  |
-------------
  b - 2a
 3b + 18
= --------
  b - 2a

which leads to the same reasoning.
.......................................
Another:

These are equations of lines, mon ami.

They have slopes, no?

You know how to find the slope of a line, oui?

If the slopes are:

--  different, then these lines will intersect in a point.

--  the same, then either:
   -- these are parallel lines
   -- these are the same line.

Try a few numbers for a,b and see what happens.  

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