(a) Consider the following system of linear equations:
x+ 5y+ z= 0
x+ 6y- z= 0
2x +ay +bz= c
Find values of a, b, and c such that the above system of linear equations has:
Add the first and second equations:
Subtract the first equation from the second:
So from the first two equations, if there are any solutions, they can only be
Substitute in the 3rd given equation:
Let's answer (ii) first.
(ii) an infinite number of solution;
If -22+2a+b=0 and 2c=0 then this equation becomes
, which has infinitely many solutions for y
That is when 2a+b=22 and c=0, for instance when a=7, b=8, c=0,
there are infinitely many solutions.
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If we solve for y
Substituting in
Let's answer (iii)
(iii) no solution
That's when y is undefined.
So that is when the denominator -22+2a+b equals 0 and the numerator is
not equal to 0, for instance when a=7, b=8, c=1, there are no solutions.
Finally we answer (i)
(i) exactly one solution;
There will be only 1 solution when -22+2a+b is not 0
For instance when a=4 and b=5 and c=1.
Edwin