Question 445381
<pre>
What are the solutions to

6x+3y+z=19
x-3y+2z=21
17x-2y+3z=86

 6x + 3y +   z =  19 ----- eq (i)
  x - 3y +  2z =  21 ----- eq (ii)
17x - 2y +  3z =  86 ----- eq (iii)
      7x +  3z =  40 ----- Adding eqs (i) & (ii) --- eq (iv)
 2x - 6y +  4z =  42 -- Multiplying eq (ii) by 2 --- eq (v)
51x - 6y +  9z = 258 -- Multiplying eq (iii) by 3 -- eq (vi)
     49x +  5z = 216 ---- Subtracting eq (v) from eq (vi) ----- (vii)
     49x + 21z = 280 ---- Multiplying eq (iv) by 7 --- eq (viii)
           16z =  64 ---- Subtracting eq (vii) from eq (viii) 
            {{{highlight(z) = 64/16 = highlight(4)}}}

     7x +  3z  = 40 ---- eq (iv)
     7x + 3(4) = 40 ---- Substituting 4 for z in eq (iv)
       7x + 12 = 40
            7x = 28
            {{{highlight(x) = 28/7 = highlight(4)}}}

x - 3y +  2z   = 21 ----- eq (ii)
4 - 3y +  2(4) = 21 ----- eq (ii) --- Substituting 4 for x, and 4 for z in eq (ii)
    4 - 3y + 8 = 21
     - 3y + 12 = 21
          - 3y = 9
            {{{highlight(y) = 9/(- 3) = highlight(- 3)}}}
 
           {{{matrix(3,1, highlight(x) = highlight(4), highlight(y) = highlight(- 3), 
highlight(z) = highlight(4))}}}</pre>