Question 207711

PROBLEM A

GIVEN: 
 X + 2Y + Z = 6  --> {{{Equation 1}}}
      X + Y = 4  --> {{{Equation 2}}}
 3X + Y + Z = 8  --> {{{Equation 3}}}


REQUIRED: FIND X, Y AND Z


SOLUTION:
FROM {{{Equation 2}}}:
X + Y = 4
    Y = 4-X --> {{{Equation 4}}}

MULTIPLY {{{Equation 1}}} WITH -1 AND ADD TO {{{Equation 3}}} TO ELIMINATE {{{Z}}}:

    -X -2Y - Z = -6  --> {{{Equation 5}}}

ADD {{{Equation 5}}} AND {{{Equation 3}}}:

    -X -2Y - Z = -6  --> {{{Equation 5}}}
    3X + Y + Z =  8  --> {{{Equation 3}}}
    _______________
    2X -Y      =  2 


         2X -Y = 2   --> {{{Equation 6}}}


NOW WE CAN SOLVE FOR {{{X}}} USING {{{Equation 4}}} AND {{{Equation 6}}} BY SUBSTITUTION:

         2X -Y = 2  --> {{{Equation 6}}}
     2X -(4-X) = 2
     2X -4 + X = 2
            3X = 2 + 4
            3X = 6
             X = 6/3
             {{{X = 2}}}

NOW THAT WE HAVE {{{X}}}, WE CAN SOLVE FOR {{{Y}}} USING {{{Equation 2}}} AND {{{X=2}}} BY SUBSTITUTION: 

         X + Y = 4  --> {{{Equation 2}}}  
         2 + Y = 4
             Y = 4-2
             {{{Y = 2}}}

NOW THAT WE HAVE {{{x}}} AND {{{Y}}}, WE CAN SOLVE FOR {{{Z}}} USING {{{Equation 1}}} BY SUBSTITUTION:

    X + 2Y + Z = 6  --> {{{Equation 1}}}
  2 + 2(2) + Z = 6  
     2 + 4 + Z = 6
         6 + Z = 6
             Z = 6-6
             {{{Z = 0}}}

{{{ANSWER TO PROBLEM A}}}  

{{{X = 2}}}
{{{Y = 2}}}
{{{Z = O}}}










PROBLEM B.

GIVEN:

    10X + Y + Z = 12  --> {{{Equation 1}}}
    8X + 2Y + Z = 11  --> {{{Equation 2}}}
 20X - 10Y - 2Z = 8   --> {{{Equation 3}}}


REQUIRED: FIND X, Y AND Z

   
SOLUTION:
MULTIPLY {{{Equation 1}}} WITH -1 AND ADD TO {{{Equation 2}}} TO ELIMINATE {{{Z}}}:

  -10X - Y -Z = -12 --> {{{Equation 1}}}*(-1)
    8X +2Y +Z =  11 --> {{{Equation 2}}} 
  _________________
  -2X  + Y    =  -1


      -2X + Y = -1  --> {{{Equation 4}}}


FROM {{{Equation 4}}}:

      -2X + Y = -1  --> {{{Equation 4}}}
            Y = 2X-1 --> {{{Equation 5}}}


NOW MULTIPLY {{{Equation 2}}} WITH {{{2}}} AND ADD TO {{{Equation 3}}} TO ELIMINATE {{{Z}}}:

    16X +  4Y + 2Z = 22 --> {{{Equation 2}}}*(2)
    20X - 10Y - 2Z = 8  --> {{{Equation 3}}}
   ___________________
    36X -  6Y      = 30
  
          36X - 6Y = 30 --> {{{Equation 6}}}


NOW WE CAN SOLVE FOR {{{X}}} USING {{{Equation 5}}} IN {{{Equation 6}}}:


          36X - 6Y = 30 --> {{{Equation 6}}}
     36X - 6(2X-1) = 30 
      36X -12X + 6 = 30
               24X = 30-6 
               24X = 24
                 X = 24/24
                 {{{X = 1}}}

NOW THAT HAVE HAVE {{{X}}}, WE CAN SOLVE FOR {{{Y}}} USING {{{Equation 5}}}:

                 Y = 2X -1 --> {{{Equation 5}}}
                 Y = 2(1)-1
                 Y = 2 - 1
                 {{{Y = 1}}}

NOW THAT HAVE HAVE {{{X}}} AND {{{Y}}}, WE CAN SOLVE FOR {{{Z}}} USING {{{Equation 1}}}, {{{Equation 2}}} OR {{{Equation 

3}}}, LETS TRY {{{Equation 1}}}:

       10X + Y + Z = 12  --> {{{Equation 1}}}
     10(1) + 1 + Z = 12
       10  + 1 + Z = 12
            11 + Z = 12
                 Z = 12-11
                 {{{Z = 1}}}


{{{ANSWER TO PROBLEM B}}}

{{{X = 1}}}
{{{Y = 1}}}
{{{Z = 1}}}











PROBLEM C 

GIVEN:

22X + 5Y + 7Z  = 12 --> {{{Equation 1}}}
10X + 3Y + 2Z  = 5  --> {{{Equation 2}}}
 9X + 2Y + 12Z = 14 --> {{{Equation 3}}}


REQUIRED: FIND X, Y AND Z


SOLUTION:

MULTIPLY {{{Equation 1}}} WITH {{{-1}}} AND ADD TO {{{Equation 2}}} AND {{{Equation 3}}} TO ELIMINATE {{{Y}}}:


 -22X - 5Y - 7Z  = -12  --> {{{Equation 1}}}*(-1)
  10X + 3Y + 2Z  =   5  --> {{{Equation 2}}}
   9X + 2Y + 12Z =  14  --> {{{Equation 3}}}
  ____________________
  -3X      + 7Z  =   7

       -3X + 7Z  =  7   --> {{{Equation 4}}}


FROM {{{Equation 4}}}:


       -3X + 7Z  =  7   --> {{{Equation 4}}}
              3X = 7Z - 7   
               X = (7Z -7)/3  --> {{{Equation 5}}}


MULTIPLY {{{Equation 3}}} WITH {{{-(3/2)}}} AND ADD TO {{{Equation 2}}} TO ELIMINATE {{{Y}}}:

 
 -(27/2)x  - 3Y  - 18z  = -21  --> {{{Equation 3}}} * (-3/2) 
      10X  + 3Y  +  2Z  =   5  --> {{{Equation 2}}}
   _________________________
     
  (-7/2)x        - 16z  = -16 


         (-7/2)x - 16z  = -16 --> {{{Equation 7}}}

FROM {{{Equation 7}}}:


        [(-7/2)x - 16z  =  -16] * 2 --> {{{Equation 7}}}*2
              -7X - 32Z = -32  --> {{{Equation 8}}}


NOW WE CAN SOLVE FOR {{{X}}} USING {{{Equation 5}}} IN {{{Equation 8}}}:

               7X - 32Z = -32  --> {{{Equation 8}}}
     -7[(7Z-7)/3] - 32Z = -32
   [(-49Z + 49)/3 - 32Z = -32] *3 
        -49Z + 49 - 96Z = -96
                  -145Z = -145
                      Z = -145/-145
                      {{{Z = 1}}}

NOW THAT WE HAVE {{{Z}}} WE CAN SOLVE FOR {{{X}}} USING {{{Equation 5}}}:

                      X = (7Z -7)/3  --> {{{Equation 5}}}
                      X = [7(1) - 7]/3
                      X = (7-7)/3
                      X = 0/3
                      {{{X = 0}}}


NOW THAT HAVE HAVE {{{Z}}} AND {{{Y}}}, WE CAN SOLVE FOR {{{Y}}} USING {{{Equation 1}}}, {{{Equation 2}}} OR {{{Equation 

3}}}, LETS TRY {{{Equation 1}}}:

         22X + 5Y + 7Z  = 12 --> {{{Equation 1}}}
      22(0) + 5Y + 7(1) = 12
             0 + 5Y + 7 = 12
                     5Y = 12-7
                     5Y = 5
                      Y = 5/5
                      {{{Y = 1}}}

{{{ANSWER TO PROBLEM C}}}

{{{X = 0}}}
{{{Y = 1}}}
{{{Z = 1}}}