Some useful resources:

But back to answering your question...

The variables:

```
sage: x, y, z, w = SR.var('x, y, z, w')
```

The equations (writing `== 0`

is not needed in order to solve):

```
sage: eqq = [x + 2*y + 2*z + 2*w, 2*x + 4*y + 6*z + 8*w, 3*x + 6*y + 8*z + 10*w]
```

The solutions (tip: naming things you compute makes them easier to reuse):

```
sage: sols = solve(eqq, x, y, z, w)
sage: sols
[[x == 2*r1 - 2*r2, y == r2, z == -2*r1, w == r1]]
```

Get the desired `V`

with:

```
sage: sol = sols[0]
sage: sol
[x == 2*r1 - 2*r2, y == r2, z == -2*r1, w == r1]
sage: V = [s.rhs() for s in sol]
sage: V
[2*r1 - 2*r2, r2, -2*r1, r1]
```

or directly:

```
sage: V = [s.rhs() for s in sols[0]]; V
[2*r1 - 2*r2, r2, -2*r1, r1]
```

The dimension of the space of solutions is the dimension of the kernel of the corresponding matrix.

Compute the matrix:

```
sage: vv = x, y, z, w
sage: a = matrix([[eq.coefficient(v) for v in vv] for eq in eqq])
sage: a
[ 1 2 2 2]
[ 2 4 6 8]
[ 3 6 8 10]
```

The kernel:

```
sage: E = a.right_kernel()
sage: E
Vector space of degree 4 and dimension 2 over Symbolic Ring
Basis matrix:
[ 1 0 -1 1/2]
[ 0 1 -2 1]
```

Dimension of the (right) kernel:

```
sage: k = E.dimension()
sage: k
2
```

Basis of the kernel:

```
sage: B = E.basis()
sage: B
[
(1, 0, -1, 1/2),
(0, 1, -2, 1)
]
```

Linear combination of kernel basis vectors, similar to the one given by `solve`

:

```
sage: s, t = SR.var('s t')
sage: E.linear_combination_of_basis((2*(s - t), t))
(2*s - 2*t, t, -2*s, s)
```

Welcome to Ask Sage! Thank you for your question!

Ideally, provide the code for entering the system into Sage, so other can copy-paste and get started.

This makes it easier for others to study your question.

Thus it increases the likelihood and speed of getting an answer.

Can you edit your question to do that?

Thank you, I just added.