Mohamed Mooney

Answered

2022-06-28

Consider the system of linear equations:

$\begin{array}{r}\{\begin{array}{l}x+ay=1\\ bx+5y=2,\end{array}\end{array}$

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.

$\begin{array}{r}\{\begin{array}{l}x+ay=1\\ bx+5y=2,\end{array}\end{array}$

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.

Answer & Explanation

laure6237ma

Expert

2022-06-29Added 27 answers

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) $ab=5\wedge \mathrm{\neg}(a=\frac{5}{2}\wedge b=2)$

The determinant is zero, but the two lines are parallel. There are no solutions

The determinant $\Rightarrow (5-ab)$

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) $ab=5\wedge \mathrm{\neg}(a=\frac{5}{2}\wedge b=2)$

The determinant is zero, but the two lines are parallel. There are no solutions

The determinant $\Rightarrow (5-ab)$

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