For a real parameter a, consider x+y+z=3, x+2y+3z=5, x+2y+(a2-1)z=a+3. - Show that every solution sa

Question

For a real parameter a, consider x+y+z=3, x+2y+3z=5, x+2y+(a2-1)z=a+3. - Show that every solution satisfies (a2-4)z=a-2. - Classify the system as having a unique solution, no solution or infinitely many solutions for every real value of a. - Give the unique solution when it exists and the complete solution family in the dependent case.

Accepted Answer

Unique for $a\ne\pm2$: $(x,y,z)=\left(\frac{a+3}{a+2},\frac{2(a+1)}{a+2},\frac1{a+2}\right)$. No solution for $a=-2$. For $a=2$: $(1+t,2-2t,t)$, $t\in\mathbb R$.

Parameters and Exceptional Cases

Parameters and Exceptional Cases

Solution:

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