SOLUTION: Solve the system of equations. 7x + 7y + z = 1 x + 8y + 8z = 8 9x + y + 9z = 9

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Question 44504: Solve the system of equations.
7x + 7y + z = 1
x + 8y + 8z = 8
9x + y + 9z = 9
Answer by AnlytcPhil(1806)   (Show Source): You can put this solution on YOUR website!
7x + 7y +  z = 1 
 x + 8y + 8z = 8 
9x +  y + 9z = 9 

1. Pick two equations and pick a letter to eliminate.
2. Eliminate that letter from the two equations you picked.
3. Pick two other equations and eliminate the SAME LETTER that
   you eliminated in step 2.
4. Now you have two equations in two unknowns.
5. Solve that system in two unknowns. You will have values for
   two of the three unknowns.
6. Substitute these values into one of the original equations
   to find the third letter, the one you eliminated in step 1.

1. Pick two equations and pick a letter to eliminate.

7x + 7y +  z = 1 
 x + 8y + 8z = 8 
9x +  y + 9z = 9

I will arbitrarily pick the first and third equations and the 
letter y to eliminate. 

2. Eliminate that letter from the two equations you picked.

 Multiply the first by -1 and the second by
7:

-1[7x + 7y +  z = 1]  
 7[9x +  y + 9z = 9]

-7x - 7y -   z = -1
56x + 7y + 63z = 63
-------------------
49x      + 62z = 62

3. Pick two other equations and eliminate the SAME LETTER that
   you eliminated in step 2.


I will now pick the first and second equations.  I must 
eliminate the same letter y to eliminate.  Multiply the first
by -1 and the second by 8:

 x + 8y + 8z = 8 
9x +  y + 9z = 9

-1[ x + 8y + 8z = 8]  
 8[9x +  y + 9z = 9]

 -x - 8y -  8z = -8
72x + 8y + 72z = 72
-------------------
71x      + 64z = 64

4. Now you have two equations in two unknowns.

49x + 62z = 62
71x + 64z = 64

5. Solve that system in two unknowns. 

Eliminate x by multiplying the first equation by
-71 and the seciond equation by 49

-71[49x + 62z = 62]
 49[71x + 64z = 64]

-3479x - 4402z = -4402
 3479x + 3136z =  3136
----------------------
        -1266z = -1266
             z = 1

Substitute z = 1 into 49x + 62z = 62

     49x + 62z = 62
   49x + 62(1) = 62
      49x + 62 = 62
           49x =  0
             x =  0

  You now have values for
   two of the three unknowns.
6. Substitute these values into one of the original equations
   to find the third letter, the one you eliminated in step 1.

Substitute z = 1 and x = 0 into the third original equation

        9x +  y + 9z = 9
     9(1) + y + 9(0) = 9
           9 + y + 0 = 9
               9 + y = 9
                   y = 0

Solution: (x, y, z) = (0, 0, 1) 
              
Edwin
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