Find a system of 2 linear equations with 3 unknowns with sol

Find a system of 2 linear equations with 3 unknowns, with solution set, the set of all scalar multiples of <1,0,1>

Solution

Let, x,y,z be the three variables

Solution is scalar multiples of

<1,0,1>

So second coordinates is always 0

ie y=0 is one equation

And first and third coordinate are equal so

x=z is another equation

So, x-z=0

y=0

So one can have

x-z+y=0

x-z-y=0

These two equations wil also give: x=z and y=0