### Instruction: Consider the system of linear equations,

2x + 3y - z = 5

x - 2y + 4z = -7 

3x + y - z = 1

Use Jacobi method to solve the above system of equations for x, y and z, starting with initial guesses x(0) = 0, y(0) = 0 and z(0) = 0. Perform 4 iterations and round your answer to 3 decimal places.### Response: The Jacobi method is an iterative method used to solve systems of linear equations. To apply the Jacobi method, we first rewrite the given system of equations in the form x = Dx + b, where D is a diagonal matrix, x is the vector of variables, and b is a constant vector. In this case, we have:

x = (1/2)(5 - 3y + z)
y = (1/3)(-5 + 2x + z)
z = (1/4)(7 + x - 2y)

Now, we will perform 4 iterations using the initial guesses x(0) = 0, y(0) = 0, and z(0) = 0.

Iteration 1:
x(1) = (1/2)(5 - 3*0 + 0) = 2.5
y(1) = (1/3)(-5 + 2*0 + 0) = -1.667
z(1) = (1/4)(7 + 0 - 2*0) = 1.75

Iteration 2:
x(2) = (1/2)(5 - 3*(-1.667) + 1.75) = 3.917
y(2) = (1/3)(-5 + 2*2.5 + 1.75) = 0.583
z(2) = (1/4)(7 + 2.5 - 2*(-1.667)) = 2.542

Iteration 3:
x(3) = (1/2)(5 - 3*0.583 + 2.542) = 3.104
y(3) = (1/3)(-5 + 2*3.917 + 2.542) = 1.151
z(3) = (1/4)(7 + 3.104 - 2*0.583) = 2.130

Iteration 4:
x(4) = (1/2)(5 - 3*1.151 + 2.130) = 2.914
y(4) = (1/3)(-5 + 2*3.104 + 2.130) = 1.046
z(4) = (1/4)(7 + 2.914 - 2*1.151) = 1.881

After 4 iterations, the approximate solutions are:
x ≈ 2.914
y ≈ 1.046
z ≈ 1.881

These values are rounded to three decimal places.