Solved by gauss elimination

3x+4y+5z=18
2x-y+8z=13
5x-2y+7x=20

There is an excellent G-J calculator here:

http://www.gregthatcher.com/Mathematics/GaussJordan.aspx

You can set the size of the matrix and enter the coefficients. Then it will show all the intermediate steps of the solution.

@Steve, that calculator doesn't work anymore

To solve this system of equations using Gauss elimination, we need to perform row operations to transform the system into an equivalent one that has a triangular or diagonal form. Here are the steps to solve it:

Step 1: Write the augmented matrix for the system of equations:
```
[ 3 4 5 | 18 ]
[ 2 -1 8 | 13 ]
[ 5 -2 7 | 20 ]
```

Step 2: Multiply rows as needed to create zeros below the pivot element (the first non-zero number in each row). To do this, we can start with the first row and eliminate the x coefficient in the second and third rows.

R2 = R2 - (2/3) * R1
R3 = R3 - (5/3) * R1

The augmented matrix becomes:
```
[ 3 4 5 | 18 ]
[ 0 -11/3 22/3 | 7/3 ]
[ 0 -14 -8 | -14 ]
```

Step 3: Next, we want to eliminate the y coefficient in the third row, so we perform the following row operation:

R3 = R3 - (14/11) * R2

The augmented matrix becomes:
```
[ 3 4 5 | 18 ]
[ 0 -11/3 22/3 | 7/3 ]
[ 0 0 8/11 | -56/11 ]
```

Step 4: Now, we have a system of equations in triangular form. We can work backwards, starting from the bottom, to solve for z, y, and x.

Equation 3: (8/11)z = -56/11
Solving for z: z = (-56/11) / (8/11) = -7

Equation 2: (-11/3)y + (22/3)(-7) = 7/3
Simplifying: (-11/3)y - 154/3 = 7/3
Solving for y: y = (7/3 + 154/3) / (-11/3) = -17/11

Equation 1: 3x + 4(-17/11) + 5(-7) = 18
Simplifying: 3x - 68/11 - 35 = 18
Combining like terms: 3x - 68/11 - 385/11 = 18
Solving for x: x = (68/11 + 385/11 - 18) / 3 = 21/11

The solution to the system of equations is x = 21/11, y = -17/11, and z = -7.