`solve()`

is a symbolic solve function, which works over the symbolic ring `SR`

not polynomial rings. Since you work with polynomials, it's better to employ the polynomial ideal machinery instead:

```
F = GF(31)
R.<x, y, z> = PolynomialRing(F)
system = [
x - (3 * y),
z - (2 * x)
]
J = Ideal(system)
print( J.variety() )
```

However, in this case you'd get an error:

ValueError: The dimension of the ideal is 1, but it should be 0

which tells us about the system having too many "degrees of freedom". Since we work in a finite field, it is easy to deal with it by fixing one of the variables, say, `x`

:

```
sols = sum( (Ideal(system+[x-x0]).variety() for x0 in F), [] )
print(sols)
```

which prints the solutions:

```
[{z: 0, y: 0, x: 0}, {z: 2, y: 21, x: 1}, {z: 4, y: 11, x: 2}, {z: 6, y: 1, x: 3}, {z: 8, y: 22, x: 4}, {z: 10, y: 12, x: 5}, {z: 12, y: 2, x: 6}, {z: 14, y: 23, x: 7}, {z: 16, y: 13, x: 8}, {z: 18, y: 3, x: 9}, {z: 20, y: 24, x: 10}, {z: 22, y: 14, x: 11}, {z: 24, y: 4, x: 12}, {z: 26, y: 25, x: 13}, {z: 28, y: 15, x: 14}, {z: 30, y: 5, x: 15}, {z: 1, y: 26, x: 16}, {z: 3, y: 16, x: 17}, {z: 5, y: 6, x: 18}, {z: 7, y: 27, x: 19}, {z: 9, y: 17, x: 20}, {z: 11, y: 7, x: 21}, {z: 13, y: 28, x: 22}, {z: 15, y: 18, x: 23}, {z: 17, y: 8, x: 24}, {z: 19, y: 29, x: 25}, {z: 21, y: 19, x: 26}, {z: 23, y: 9, x: 27}, {z: 25, y: 30, x: 28}, {z: 27, y: 20, x: 29}, {z: 29, y: 10, x: 30}]
```

**ADDED.** As John pointed out, solutions of the form like `x = 3 * y, z = 6 * y`

can be seen from the generators of the ideal:

```
print(Ideal(system).gens())
```

which gives

```
[x - 3*y, -2*x + z]
```

that is, `y = x/2`

and `z = 2*x`

.