Solve the following equations using Cramer's Rule: 5x-4y+6z=58 -4x+6y+3z=-13 6x+3y+7z=53

We can apply Cramer's rule if and only if the determiant
of the system is different from zero.

We'll calculate the
determinant:

det A = 5*6*7 + (-4)*3*6 + 3*6*(-4) - 6*6*6 -
3*3*5 - 7*16

det A = 210 - 2*4*18 - 216 - 45 -
112

det A = 210 - 517

det A =
-307

Since det A is different from zero, we'll apply
Cramer's rule:

x = det X/detA, y = detY/detA, z =
detZ/detA

5x-4y+6z=58

-4x+6y+3z=-13

6x+3y+7z=53

detX
= 58*6*7 - 13*18 - 12*53 - 53*36 - 9*58 + 13*28

detX = 2436
- 234 - 636 - 1908 - 522 + 364

det x =
-1228

x =
-1228/-307

x =
4

y =
614/-307

y =
-2

In the same way, we'll determine det y
and det z, substituting the column of the coefficients of the variable taht has to be
determined, by the column of the coefficients of the right
side.

Now, we'll substitute x and y in the first
equation:

20 + 8 + 6z = 58

6z
= 58 - 28

6z =
30

z =
5

The solution of the system
is {4 ; -2 ; 5}.