If for some αandβin R , the system of equations x+4y−2z=1,x+7y−5z=β,x+5y+αz=5has many solutions , then 2α+β=

The condition for a non-homogeneous system of equations to have infinitely many solutions is Δ=Δ1=Δ2=Δ3=0.  NowΔ=0⇒14-217-515α=0⇒17α+25-4α+5-25-7=0⇒3α=-9⇒α=-3Also Δ3=0⇒14117β155=0⇒135-5β-45-β+15-7=0⇒13-β=0⇒β=13∴2α+β=7