Question

What is the solution to the system (1)x-y-2z=4(2)-x+2y+z=1(3)-x+y-3z=11

Answer 1

Step 1:

Write the system of equations

x - y - 2z = 4 (1)

-x + 2y + z = 1 (2)

-x + y - 3z = 11 (3)

Step 2:

From equation 1, make x subject of the relation and substitute in equation 2 to get equation 4 and equation 3 to get equation 5.

`\begin{gathered} \text{From (1)} \\ x\text{ - y - 2z = 4} \\ x\text{ = 4 + y + 2z} \\ In\text{ (2) -x + 2y + z = 1} \\ -(4\text{ + y + 2z) + 2y + z = 1} \\ -4\text{ - y - 2z + 2y + z = 1} \\ y\text{ - z = 1 + 4} \\ y\text{ - z = 5 (4)} \\ In\text{ (2) -x + y - 3z = 11} \\ -(4\text{ + y + 2z) + y - 3z = 11} \\ -4\text{ - y - 2z + y - 3z = 11} \\ \text{Collect similar terms.} \\ -5z\text{ = 11 + 4} \\ -5z\text{ = 15 (5)} \\ \text{z = }(15)/(-5) \\ z\text{ = -3} \end{gathered}`

Step 3:

From equation 4, find the value of y.

`\begin{gathered} y\text{ - z = 5} \\ y\text{ - (-3) = 5} \\ y\text{ + 3 = 5} \\ y\text{ = 5 - 3} \\ y\text{ = 2} \end{gathered}`

Step 4:

Substitute y and z in x = 4 + y + 2z to find the value of x.

`\begin{gathered} x\text{ = 4 + y + 2z} \\ x\text{ = 4 + 2 + 2(-3)} \\ \text{x = 4 + 2 - 6} \\ x\text{ = 0} \end{gathered}`

Final answer

x = 0

y = 2

z = -3