If E is a space of linear equations in 3 variables with standard operations, how can we show that W is a...

subspace of E, if the equations have x=1, y= -1 and z=3 as solutions.

W = { ax + by + cz = d | a,b,c,d elements of real numbers and x=1, y= -1 and z=3 }

Note that W can have other solutions as well.

Thanks, please show steps and explain if possible.

I am going to use vector '*' dot-products here,
Let X =(x,y,z) a member of R3, d member of R, let A =(a,b,c) also in R3:

Let E = {space spanned by X | (x,y,z)*(a,b,c) =d}
W = (space spanned by A |(a,b,c)*(1,-1,3)=d}
dot-product is commutative =>
W = (space spanned by A | (1,-1,3)*(a,b,c)=d}

since X = (1,-1,3) is also in E
then W is sub-space E. QED