# Thread: Need some help with system of equations

1. ## Need some help with system of equations

Sorry if this is the wrong sub-forum to post this on, can a admin move if wrong?

need some help with this system of equations

find the value of the parameters a and p for which the system

x + 5y = 6,
6y + az = a + 1 - p,
x - ay + z = p + 2

i. unique solution
ii. infinitly many solutions
iii. no solutions

2. ## Re: Need some help with system of equations

go to the link ... scroll down to topic #3 Cramer's Rule - system of three linear equations in three variables

Solving systems via Cramer's Rule - CATs

3. ## Re: Need some help with system of equations

One way to do this is to try to solve the system. From x+ 5y= 6, x= 6- 5y so x - ay + z = p + 2 becomes 6- 5y- ay+ z= 6- (a+ 5)y+ z= p+ 2 or (a+ 5)y+ z= p- 4. Then z= p- 4- (a+ 5)y. Putting that into 6y+ az= a+ p- 1, we have 6y+ a(p- 4- (a+ 5)y)= 6y- (a^2+ 5a)y+ap- 4a= a+ p- 1 or (a^2+ 5a- 6)y= (a+ 6)(a- 1)y= a+ p- 1- ap+ 4a.

Now we would like to solve that for y by dividing both sides by (a+ 6)(a- 1). If neither a+ 6 or a- 1 is 0, then we can divide by (a+ 6)(a- 1) and have a unique solution. If is either -6 or 1, then (a+ 6)(a- 1) is 0 the equation becomes 0= a+ p- 1- ap+ 4a. There is no y in that equation but there are still two possibilities. If a+ p- 1+ ap+ 4a is NOT 0 then the equation is untrue for any y. But f a+ p- 1+ ap+ 4a is 0, then the equation is 0 which is always true, for all y.