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Question

Solve the equations
x+ay+a2z=a3,x+by+b2z=b3,x+cy+c2z=c3.

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Solution

Δ=∣ ∣ ∣1aa21bb21cc2∣ ∣ ∣=(ab)(bc)(ca)

Δ1=∣ ∣ ∣a3aa2b3bb2c3cc2∣ ∣ ∣=abc(ab)(bc)(ca)

Δ2=∣ ∣ ∣1a3a21b3b21c3c2∣ ∣ ∣=(ab)(bc)(ca)(ab+bc+ca)

Δ3=∣ ∣ ∣1aa31bb31cc3∣ ∣ ∣=(ab)(bc)(ca)(a+b+c)

x=Δ1Δ=abc;y=Δ2Δ=(ab+bc+ca);z=Δ3Δ=(a+b+c)


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