Question 208209

SOLUTION:

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

                |  1  -1  5 |  2 |
                |  4  -3  5 |  3 |
                |  3  -2  4 |  1 |


By Cramers Rule:  

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


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


D =  |  1  -1   5 |  
     |  4  -3   5 |  
     |  3  -2   4 |  

D = 1(-12+10) - 4(-4+10) + 3(-5+15)
D = 1(-2) -4(6) + 3(10)
D = -2 -24 +30
D = -26 +30

{{{D = 4}}}


Dx = |  2  -1   5 |  
     |  3  -3   5 |  
     |  1  -2   4 |

Dx = 2(-12+10) - 3(-4+10) +1(-5+15)
Dx = 2(-2) - 3(6) + 1(10)
Dx = -4 - 18 + 10
Dx = -22 +10

{{{Dx = -12}}}


Dy =  |  1   2   5 |  
      |  4   3   5 |  
      |  3   1   4 | 

Dy = 1(12-5) -4(8-5) +3(10-15)
Dy = 1(7) -4(3) +3(-5)
Dy = 7 -12 -15
Dy = -5 -15

{{{Dy = -20}}}


Dz = |  1  -1   2 |  
     |  4  -3   3 |  
     |  3  -2   1 |

Dz = 1(-3+6)- 4(-1+4)+ 3(-3+6)
Dz = 1(3) - 4(3) + 3(3)
Dz = 3 -12 + 9
Dz = -9 + 9

{{{Dz = 0}}}



{{{x = Dx/D = -12/4}}} 
{{{x = -3}}}

{{{y = Dy/D = -20/4}}} 
{{{y = -5}}}

{{{z = Dz/D = 0/4}}} 
{{{z = 0}}}


Answer: The solution set is {(-3,-5, 0)}