Final answer:
The solution to the system of equations is x = 27/5, y = 2, z = 12/5.
Step-by-step explanation:
To find the solution to the given system of equations:
2x - y + z = 4
4x - 2y + 2z = 8
-x + 3y - z = 5
We can use the method of solving simultaneous equations by elimination or substitution. Let's solve it using the elimination method:
First, multiply the third equation by 2 to match the coefficients of z.
-2x + 6y - 2z = 10
Now, add the first and second equations together.
2x - y + z + 4x - 2y + 2z = 4 + 8
6x - 3y + 3z = 12
Next, add this new equation to the multiplied third equation.
6x - 3y + 3z + -2x + 6y - 2z = 12 + 10
4x + 3y + z = 22
Simplify the equation:
4x + 3y + z = 22
Now, we have a system of two equations:
4x + 3y + z = 22
-x + 3y - z = 5
With this system, we can use elimination method or substitution method to find the values of x, y, and z.
Solving for x, we have:
5x = 27
x = 27/5
Solving for y, we have:
3y = 6
y = 6/3
Solving for z, we have:
z = 5 + x - 3y
Substitute the values of x and y:
z = 5 + (27/5) - 3(6/3)
Simplify the equation:
z = 5 + 27/5 - 18/3
z = 15/5 + 27/5 - 30/5
Simplify further:
z = 12/5
Therefore, the solution to the system of equations is:
x = 27/5, y = 2, z = 12/5