Question 208208
REQUIRED: Find the solution set {(x, y z)} for the system given.

SOLUTION:

The augmented 3x3 matrix for the above system of equations is:

                |  4  -2  -3 |  8 |
                |  5   3  -4 |  4 |
                |  6  -4  -5 | 12 |

By Cramers Rule:  

{{{x = Dx/D}}}
{{{y = Dy/D}}}
{{{z = Dz/D}}}


First, lets solve for D, Dx, Dy and Dz:


D =  |  4  -2  -3 | 
     |  5   3  -4 | 
     |  6  -4  -5 | 

D = 4(-15-16) -5(10-12) +6(8+9)
D = 4(-31) -5(-2)+ 6(17)
D = -124 +10 +102
D = -124 -112

{{{D = -12}}}


Dx = |  8  -2  -3 | 
     |  4   3  -4 | 
     | 12  -4  -5 |

Dx = 8(-15-16) -4(10-12) + 12(8+9)
Dx = 8(-31) -4(-2) + 12(17)
Dx = -248 +8 + 204
Dx = -248 +212

{{{Dx = -36}}}


Dy = |  4   8  -3 | 
     |  5   4  -4 | 
     |  6  12  -5 | 


Dy = 4(-20+48) -5(-40+36) +6(-32+12)
Dy = 4(28) -5(-4) +6(-20)
Dy = 112 +20 -120
Dy = 132 -120

{{{Dy = 12}}}


Dz = |  4  -2   8 | 
     |  5   3   4 |  
     |  6  -4  12 |

Dz = 4(36+16) -5(-24+32) +6(-8-24)
Dz = 4(52) -5(8) +6(-32)
Dz = 208 -40 -192
Dz = 168 -192

{{{Dz = -24}}}



{{{x = Dx/D = -36/-12}}} 
{{{x = 3}}}

{{{y = Dy/D = 12/-12}}} 
{{{y = -1}}}

{{{z = Dz/D = -24/-12}}} 
{{{z = 2}}}


Answer: The solution set is {(3,-1,2)}