Question

Solve the following system of equations.

x + 3y – z = 2
x – 2y + 3z = 7
x + 2y – 5z = –21

(2, 3, 5)

(–2, 3, 5)

(2, –3, 5)

(2, 3, –5)

Answer 1

it can be solved either by substitution or matrix method

Answer 2

pretty lengthy process

Answer 3

x + 3y - z = 2
x - 2y + 3z = 7
x + 2y - 5z = - 21
-----------------
start with the first two equations....
x + 3y - z = 2 --->(-1)x + 3y - z = 2
x - 2y + 3z = 7
---------------
-x - 3y + z = -2 (result of multiplying by -1)
x - 2y + 3z = 7
---------------add
-5y + 4z = 5

now take the last two equations and eliminate x
x - 2y + 3z = 7
x + 2y - 5z = - 21 -->(-1)x + 2y - 5z = - 21
-----------------
x - 2y + 3z = 7
-x - 2y + 5z = 21 (result of multiplying by -1)
------------------add
- 4y + 8z = 28

now add the answers you got from the first two equations and the last two equations...
-5y + 4z = 5 --> (-2)-5y + 4z = 5
-4y + 8z = 28
-------------
10y - 8z = - 10 (result of multiplying by -2)
-4y + 8z = 28
--------------add
6y = 18
y = 18/6
y = 3

now sub 3 in for y in either of the answers you got from the first two equations or the last two equations...
-4y + 8z = 28
-4(3) + 8z = 28
-12 + 8z = 28
8z = 28 + 12
8z = 40
z = 40/8
z = 5

now sub 3 in for y and 5 in for z in any of the original equations...
x + 3y - z = 2
x + 3(3) - 5 = 2
x + 9 - 5 = 2
x = 2 + 5 - 9
x = 7 - 9
x = -2

check...
x - 2y + 3z = 7
-2 - 2(3) + 3(5) = 7
-2 - 6 + 15 = 7
- 8 + 15 = 7
7 = 7 (correct)

ANSWER : x = -2, y = 3, z = 5 (-2,3,5)