Solve: x+y+ 4z = 6, 3x +2y +z =4 2x+ 2y + z =9 If the roots of the equation ax^2 + bx + c are 6 and 5 what are values of a and b?

Since the given equations of the system are all linear
equations, we'll solve the system using another
method.

We'll calculate the determinent of the system. The
determinant of the system is formed from the coefficients of the variables, x, y and
z.

We'll note the determinant as det
A.

             1   1   4

det
A =  3   2   1

              2   2  
1

We'll calculate det A:

det A
= 1*2*1 + 3*2*4  + 1*1*2 - 4*2*2 - 2*1*1 - 3*1*1

det A = 2
+ 24 + 2 - 16 - 2 - 3

We'll eliminate and combine like
terms:

det A = 7

Now, we'll
calculate the variable x using Cramer formula:

x = detX /
detA

Det X is the determinant whose column of coefficients
of the variable that has to be found (in this case x) is substituted by the column of
the terms from the right side of the equal (6 , 4 ,
9).

            6   1   4

det
X = 4   2   1

            9    2  
1

det X = 6*2*1 + 4*2*4 + 1*1*9 - 4*2*9 - 6*2*1 -
4*1*1

We'll eliminate like
terms:

detX = 32 + 9 - 72 - 12 -
4

det X = -47

x =
detX/detA

x =
-47/7

            1   6  
4

det Y = 3   4  
1

             2   9   1

det Y
= 4 + 108 + 12 - 32 - 9 - 18

y =
detY/detA

y =
65/7

We'll calculate z substituting the
values of x and y into the first equation:

x+y+ 4z =
6

4z = 6 - x - y

z = (6  -x -
y)/4

z = (6 + 47/7 - 65/7)/4

z
= -12/7*4

z =
-3/7