Mohamed Mooney

2022-06-28

Consider the system of linear equations:
$\begin{array}{r}\left\{\begin{array}{l}x+ay=1\\ bx+5y=2,\end{array}\end{array}$
where $a$ and $b$ are parameters.
(a) Determine the conditions on $a$ and $b$ to get a unique solution.
(b) Determine the conditions on $a$ and $b$ to get infinitely many solutions.
(c) Determine the conditions on $a$ and $b$ such that the system has no solutions.

laure6237ma

Expert

a) $ab\ne 5$
The determinant of the matrix not equal to zero. There is a unique solution
b) $a=\frac{5}{2},b=2$
The determinant is zero, but the two lines are identical, There are infinitely many solution
c) $ab=5\wedge \mathrm{¬}\left(a=\frac{5}{2}\wedge b=2\right)$
The determinant is zero, but the two lines are parallel. There are no solutions
The determinant $⇒\left(5-ab\right)$

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