To solve the system of equations:
- (2x + y - z = 11)
- (3x + 2y - 4z = 15)
- (4x + 3y + 7z = 19)
You can use the method of elimination or substitution. Here, I'll demonstrate using elimination:
Step 1: Eliminate (y) from equations 1 and 2.
Multiply equation 1 by 2 to align the coefficients of (y):
[
4x + 2y - 2z = 22
]
Subtract equation 2 from this new equation:
[
(4x + 2y - 2z) - (3x + 2y - 4z) = 22 - 15 \implies x + 2z = 7
]
Step 2: Eliminate (y) from equations 2 and 3.
Multiply equation 2 by 3 and equation 3 by 2:
[
9x + 6y - 12z = 45
]
[
8x + 6y + 14z = 38
]
Subtract the second equation from the first:
[
(9x + 6y - 12z) - (8x + 6y + 14z) = 45 - 38 \implies x - 26z = 7
]
Step 3: Solve the two new equations:
From the results of steps 1 and 2, we have:
- (x + 2z = 7)
- (x - 26z = 7)
Subtract the second from the first:
[
(x + 2z) - (x - 26z) = 7 - 7 \implies 28z = 0 \implies z = 0
]
Plug (z = 0) into (x + 2z = 7):
[
x + 2(0) = 7 \implies x = 7
]
Step 4: Find (y) using equation 1:
Substitute (x = 7) and (z = 0) into equation 1:
[
2(7) + y - 0 = 11 \implies 14 + y = 11 \implies y = -3
]
Solution:
(x = 7), (y = -3), (z = 0).
