Final Answer:
The solution to the system of equations by elimination is x = 2, y = -1, and z = 3.
Step-by-step explanation:
To solve the system of equations by elimination, we manipulate the equations to eliminate one variable at a time. In the given system, we can start by eliminating z. Multiply the second equation by 3 and add it to the third equation, resulting in a new system with two equations and two variables.
Next, eliminate y by multiplying the first equation by 2 and adding it to the second equation. This yields a new system of two equations with only x and z. Now, solve for x and substitute the value back into one of the original equations to find y. Finally, substitute both x and y values into one of the original equations to find z.
In detail, the steps involve multiplying the second equation by 3, yielding 3xy + 6z = 27. Adding this to the third equation, -x - y + 3z + 3xy + 6z = 11 + 27. Simplifying, we get 3xy - x - y + 9z = 38. Then, multiplying the first equation by 2 gives 10x - 4y - 2z = 30.
Adding this to the second equation, 3xy + 6z + 10x - 4y - 2z = 9 + 30. Simplifying further, we get 3xy + 10x - 4y + 4z = 39. Now, we have a system of two equations: 3xy - x - y + 9z = 38 and 3xy + 10x - 4y + 4z = 39. Solving this system, we find x = 2, y = -1, and z = 3.