if x/q+r-p=y/r+p-q=z/p+q-r,prove that(q-r)x+(r-p)y+(p-q)z

Question

it is a problem given in examples 1 of ''HigherAlgebra'a book by H.S.Hall and S.R.Knight

Answer 1

Question: if x/q+r-p=y/r+p-q=z/p+q-r,prove that(q-r)x+(r-p)y+(p-q)z =0.

Let u = (q-r)
Let v = (r-p)
Let w = (p-q)

Then x/(q+v) = y/(r+w) = z/(p+u)  =>  u.x + v.y + w.z = 0

The equation pair x/(q+v) = z/(p+u) -> xp + xu = zq + zv   ---- (1)
The equation pair x/(q+v) = y/(r+w) -> xr + xw = yq + yv   ---- (2)
The equation pair y/(r+w) = z/(p+u) -> zr + zw = yp + yu   ---- (3)

From (1),  ux = zq + zv - xp
From (2),  vy = xr + xw - yp
From (3),  wx = yp + yu - zx

Adding together the above three eqns,

ux + vy + wz = x(r+w-p) + y(p+u-q) + z(q+v-r)
ux + vy + wz = x(r - q) + y(p - r) + z(q - p)
ux + vy + wz = -ux - uv - wz
2(ux + vy + wz) = 0
ux + vy + wz = 0

i.e. (q-r).x + (r-p).y + (p-q).z = 0