Answer:
The solution is x = 3 , y = 0 , z = -5
Step-by-step explanation:
* In the system of equations we can solve by using substitution
method
- Means find the value of one variable in terms of another variable
and then substitute it in another equation
* Lets do that in our problem
∵ x + y - z = 8 ⇒ (1)
∵ 2x - 2y - z = 11 ⇒ (2)
∵ x - 5y + 3z = 12 ⇒ (3)
* Use (1) to find the value of z in terms of x and y
∵ x + y - z = 8 ⇒ subtract 8 from both sides
∴ x + y - z - 8 = 0 ⇒ add z in the both sides
∴ x + y - 8 = z ⇒ (4)
* Substitute (4) in (2)
∴ 2x - 2y - (x + y - 8) = 2x - 2y - x - y + 8 = 11
∴ x - y + 8 = 11 ⇒ subtract 8 from both sides
∴ x - y = 3 ⇒ (5)
* Substitute (4) in (3)
∴ x - 5y + 3(x + y - 8) = x - 5y + 3x + 3y - 24 = -12
∴ 4x - 2y - 24 = -12 ⇒ add 24 to both sides
∴ 4x - 2y = 12 ⇒ ÷ it by 2
∴ 2x - y = 6 ⇒ (6)
* Use (5) to find x in terms of y
∵ x - y = 3 ⇒ add y to both sides
∴ x = 3 + y ⇒ (7)
* Substitute (7) in (6)
∴ 2(3 + y) - y = 6
∴ 6 + 2y - y = 6
∴ 6 + y = 6 ⇒ subtract 6 from both sides
∴ y = 0 ⇒ substitute this value of y in (7)
∴ x = 3 + 0 = 3 ⇒ substitute the values of x and y in (4)
∴ z = 3 + 0 - 8 = -5
* The solution is x = 3 , y = 0 , z = -5
