Solve the system x+y-z=8 2x-2y-z=11 x-5y+3z=-12 using substitution

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