solve the system; x+y+2z=9 3x+5y-z=21 x+2y+3z=15?

x + y + 2z = 9 ...... [1]
3x + 5y - z = 21 ...... [2]
x + 2y + 3z = 15 ...... [3]

[1]*3 :
3x + 3y + 6z = 27 ...... [4]

[3] - [1] :
(x - x) + (2y - y) + (3z - 2z) = 15 - 9
y + z = 6 ...... [5]

[4] - [2] :
(3x - 3x) + (3y - 5y) + (6z + z) = 27 - 21
-2y + 7z = 6 ...... [6]

[5] = [6] :
y + z = -2y + 7z
3y = 6z
y = 2z ...... [7]

Substitute [7] into [5] :
2z + z = 6
3z = 6
z = 2

Substitute z = 2 into [7] :
y = 2*2
y = 4

Subsititute y = 4 and z = 2 into [1] :
x + 4 + 2*2 = 9
x = 1

Hence, x = 1, y = 4, z = 2

x + y + 2z = 9
6x + 10y - 2z = 42_______ADD

7x + 11y = 51

x + y + 2z = 9
x + 2y + 3z = 15

3x + 3y + 6z = 27
-2x - 4y - 6z = - 30________ADD
x - y = - 3

7x + 11y = 51
x - y = - 3

7x + 11y = 51
11x - 11y = - 33_____ADD

18x = 18
x = 1

x - y = - 3
1 - y = - 3
y = 4

1 + 2 + 3z = 15
z = 4

x = 1 , y = 4 , z = 4

(1) : x + y + 2z = 9

(2) : 3x + 5y - z = 21

(3) : x + 2y + 3z = 15


You calculate (3) - (1) and you can get the equation (4)

(4) : (x + 2y + 3z) - (x + y + 2z) = 15 - 9

(4) : x + 2y + 3z - x - y - 2z = 6

(4) : y + z = 6

(4) : y = 6 - z


You calculate (2) - [3 * (1)]

(3x + 5y - z) - 3.(x + y + 2z) = 21 - (3 * 9)

3x + 5y - z - 3x - 3y - 6z = 21 - 27

2y - 7z = - 6 → recall (4): y = 6 - z

2.(6 - z) - 7z = - 6

12 - 2z - 7z = - 6

- 9z = - 18

→ z = 2


Recall (4): y = 6 - z

y = 6 - 2

→ y = 4


Recall (1): x + y + 2z = 9

x = 9 - y - 2z

x = 9 - 4 - 4

→ x = 1

3x + 5y – z = 21 …………………(1)
x + 2y + 3z =15 …………………(2)
x + y + 2z = 9 ………………..(3)

(2) + (1) – (3)
3x + 6y = 27, or, dividing by 3,
x + 2y = 9 ………………………………(4) subtract (4) from (2) to isolate z
3z = 6, so z = 2, remove 2z from (3)
x + y = 5 ……………………………..(5) subtract (5) from (4)
y = 4, and so x = 1