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EEJ MAIN Mathematics QUESTION #6019
Question 1

Let $S$ be the set of distinct values of $b$ for which the linear system $x+y+z=1$, $x+ay+z=1$, $ax+by+z=0$ has no solution. Describe the set $S$.

  • An empty set
  • An infinite set
  • A finite set with two or more elements
  • A singleton✔️
Correct Answer Explanation

Subtract equation 1 from equation 2: $(a-1)y = 0$.

Case $a\neq1$: $y=0$. Then from eq.1 and eq.3: $x+z=1$ and $ax+z=0$, giving $(a-1)x=-1$, so $x=\frac{-1}{a-1}$. This always has a solution for $a\neq1$.

Case $a=1$: System becomes $x+y+z=1$, $x+y+z=1$, $x+by+z=0$. The third equation forces $-1=by-... $ i.e. subtracting eq.1: $(b-1)y=-1$. For no solution we need $b=1$.

So $S=\{1\}$ — a single value. Hence $S$ is a singleton.