Consider the system of equations $x+y=2$, $ax+y=b$.Find the conditions on $a$ and $b$ under which
(i) the system has exactly one solution;
(ii) the system has no solution;
(iii) the system has more than one solution.
I solved for $x$ and $y$ in terms of $a$ and $b$ and got some restricting conditions. But, it is difficult to solve that way. I was wondering if it could be solved like the determinants or matrices with Cramer's Rule or something similar because I read similar about similar conditions there. Please help!
Best Answer
By looking carefully at this system we can answer the questions by inspection , hardly doing any arithmetic at all.
$ \ \ x + y = 2 \\ ax + y = b $
The coefficients of y are equal therefore there are two cases.
Case 1
If a = 1 then the left hand sides of the equations are identical. If $b \ne 2 $ then the lines are parallel (not coinciding) , therefore no solution. If b = 2 then the lines (are parallel) coincide , therefore infinite number of solutions.
Case 2
If $ a \ne 1 $ then there is no way to make the left hand sides identical (parallel) because of the y coefficients therefore there will be a unique solution.
Summary
(i) Unique solution for $a \ne 1 $
(ii) No solution for $ a = 1 $ AND $ b \ne 2 $
(iii) Infinite number of solutions for $ a = 1 $ AND $ b = 2 $