To begin, we must rearrange the equations into a form where the variable on the left side is isolated:
1. 83x + 11y - 4z = 95
2. 3x + 8y + 28z = 71
3. 7x + 52y - 13z = 104
Now, we will express the equations in matrix form:
A = [[83, 11, -4],
[3, 8, 28],
[7, 52, -13]]
b = [[95], [71], [104]]
Starting with initial values of x0 = y0 = z0 = 0, and using the Gauss-Jacobi iteration formula:
x(k+1) = (b1 - a11*x(k) - a12*y(k) - a13*z(k))/a11
y(k+1) = (b2 - a21*x(k) - a22*y(k) - a23*z(k))/a22
z(k+1) = (b3 - a31*x(k) - a32*y(k) - a33*z(k))/a33
We will plug in the initial values to estimate new values for x1, y1, and z1. We will continue this process until the values converge. Let's start solving for x1:
x(1) = (95 - 83*0 - 11*0 - (-4)*0)/83
x(1) = 95/83 = 1.145
Next, we will solve for y1 and z1 using the same formula:
y(1) = (71 - 3*0 - 8*0 - 28*0)/8
y(1) = 71/8 = 8.875
z(1) = (104 - 7*0 - 52*0 - (-13)*0)/(-13)
z(1) = 104/-13 = -8
By continuing this process of iteration, we can find the values where the system converges.
Solve the following system of equations by
Gauss-jacobi method
83x + 11y - 4z = 95
3x + 8y + 28z = 71
7x + 52y - 13x 104
3 answers
Solve the following system of equations by
Gauss-jacobi method
83x + 11y - 4z = 95
3x + 8y + 28z = 71
7x + 52y - 13x 104
Gauss-jacobi method
83x + 11y - 4z = 95
3x + 8y + 28z = 71
7x + 52y - 13x 104
It seems like you made a mistake in the third equation provided. There seems to be a typographical error as it should be 7x + 52y - 13z = 104 instead of 7x + 52y - 13x = 104.
Let's write the corrected system of equations and proceed with the Gauss-Jacobi method:
1. 83x + 11y - 4z = 95
2. 3x + 8y + 28z = 71
3. 7x + 52y - 13z = 104
Now, we express the equations in matrix form:
A = [[83, 11, -4],
[3, 8, 28],
[7, 52, -13]]
b = [[95], [71], [104]]
We start with initial values of x0 = y0 = z0 = 0 and use the Gauss-Jacobi iteration formula to find updated values for x, y, and z.
After the first iteration:
x(1) = (95 - 11*0 + 4*0)/83 = 95/83 ≈ 1.145
y(1) = (71 - 3*0 - 28*0)/8 = 71/8 = 8.875
z(1) = (104 - 7*0 - 52*0)/(-13) = -104/13 ≈ -8
After further iterations, we can continue updating the values of x, y, and z until they converge to the solution of the system of equations.
Let's write the corrected system of equations and proceed with the Gauss-Jacobi method:
1. 83x + 11y - 4z = 95
2. 3x + 8y + 28z = 71
3. 7x + 52y - 13z = 104
Now, we express the equations in matrix form:
A = [[83, 11, -4],
[3, 8, 28],
[7, 52, -13]]
b = [[95], [71], [104]]
We start with initial values of x0 = y0 = z0 = 0 and use the Gauss-Jacobi iteration formula to find updated values for x, y, and z.
After the first iteration:
x(1) = (95 - 11*0 + 4*0)/83 = 95/83 ≈ 1.145
y(1) = (71 - 3*0 - 28*0)/8 = 71/8 = 8.875
z(1) = (104 - 7*0 - 52*0)/(-13) = -104/13 ≈ -8
After further iterations, we can continue updating the values of x, y, and z until they converge to the solution of the system of equations.