Hi there, I'm trying to work out this question, but not having much luck:
Solve this system of equations for all possible values of and , where and are real constants.
The question says to be careful with special values.
Thanks for your help!
I have no idea what you mean by "The only possible values of a and b that I get are both 1". a and b can have any values at all. For example if a= b= 0, the equations reduce to y+ z= 1, x+ z= 1, and z= 1 which has solution x= y= 0, z= 1.
The "special values" referred to are a= b= 1 for which the equations do not have a single value.
This question is all about making sure you never divide by something that might be zero. So, continuing from here:...(4)by to get:
Substitute into (2):
...(5)So if we can now divide both sides by to get:Substitute these values of and into (1) to get:Again, we must ensure that we don't divide by zero. So, if , we can divide both sides by and get:
...(6)Therefore, provided and and , the solution is:
Now we need to re-trace our steps, and see what happens if one or more of these conditions is not satisfied.
Starting with the last condition first, if , then equation (6) is satisfied by any value of , say . So, if and we get the solution:, for any value of .Next, if and , then (5) is satisfied by any value of , say . Substituting this value into (1) (with and ) we get:
So the solution for and is:
, for any value of .Now you need to go back and see what happens if , where equation (4) is then satisfied by any value of .