a) 2x-2y-5z=0 ; 4x-y+z=0 ; 3x-2y+3z=0 by reduction method
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If the system of equations x+y+z=5, x+2y+3z=9, x+3y+αz=β has infinitely many solutions, then β−α equals?
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D=
∣
∣
∣
∣
∣
∣
∣
∣
1
1
1
1
2
3
1
3
α
∣
∣
∣
∣
∣
∣
∣
∣
=
∣
∣
∣
∣
∣
∣
∣
∣
1
0
0
1
1
2
1
2
α−1
∣
∣
∣
∣
∣
∣
∣
∣
=(α−1)−4=(α−5)
for infinite solutions D=0⇒α=5
D
x
=0⇒
∣
∣
∣
∣
∣
∣
∣
∣
5
9
β
1
2
3
1
3
5
∣
∣
∣
∣
∣
∣
∣
∣
=0
⇒
∣
∣
∣
∣
∣
∣
∣
∣
0
−1
β−15
0
−1
−2
1
3
5
∣
∣
∣
∣
∣
∣
∣
∣
=0
⇒2+β−15=0
⇒β−13=0
on β=13 we get D
y
=D
z
=0
α=5,β=13.
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