Given three distinct positive constants $a , b , c$ we want to solve the simultaneous equations $$\begin{aligned}
a x + b y & = \sqrt { 2 } \\
b x + c y & = \sqrt { 3 }
\end{aligned}$$ (a) There exists a combination of values for $a , b , c$ such that the above system has infinitely many solutions $( x , y )$. (b) There exists a combination of values for $a , b , c$ such that the above system has exactly one solution $( x , y )$. (c) Suppose that for a combination of values for $a , b , c$, the above system has NO solution. Then $2 b < a + c$. (d) Suppose $2 b < a + c$. Then the above system has NO solution.
Given three distinct positive constants $a , b , c$ we want to solve the simultaneous equations
$$\begin{aligned}
a x + b y & = \sqrt { 2 } \\
b x + c y & = \sqrt { 3 }
\end{aligned}$$
(a) There exists a combination of values for $a , b , c$ such that the above system has infinitely many solutions $( x , y )$.\\
(b) There exists a combination of values for $a , b , c$ such that the above system has exactly one solution $( x , y )$.\\
(c) Suppose that for a combination of values for $a , b , c$, the above system has NO solution. Then $2 b < a + c$.\\
(d) Suppose $2 b < a + c$. Then the above system has NO solution.