Question

How do I solve this using the elimination method

Answer 1

(a) first equation        x + 2y - 3z  = 11
(b) second equation  2x + y - 2z  = 9
(c) third equation      4x + 3y +z  = 16
1) let's eliminate x from (a) and (b)
    multiply (a) by - 2 → -2x -4y +3z = -22 and add it to b):
-2x - 4y +6z = -22
2x  + y   -2z = 9
---------------     ----
0x -3y +4z    =  - 13   → -3y + 4z = -13 (d)

2) Now eliminate x from b) and c)
    multiply b) by -2 → -4x - 2y +4z = - 18 and add it to c):
-4x - 2y + 4z = -18
4x + 3y + z = 16
--------------    ----
0x + y + 5z = -2 → y+ 5z = -2 (e)

3) Now we have 2 equations with 2 variables:
(d) -3y + z = -13
(e)  y+ 5z = -2

4) again eliminate y from (d) and (3). by
multiplying (e) by 3 → 3y +15 z = -6 (f).

Add d) to f):
e) -3y + 4z = -13
f) 3y +15 z = -6
--------------   ----
0y  + 19z    =  - 19  and z = -1
Now replace z by - 1 , starting from the last pair of equation and up, you will find:
x = 2,  y = 3  and z = -1