Consider the system of linear equations: { x + a y = 1 b x + 5 y = 2 , 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.

Question

Consider the system of linear equations:                              {                                                    x                +                a                y                =                1                                                                    b                x                +                5                y                =                2                ,                                                                  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 · Accepted Answer

a)   a  b  ≠  5The determinant of the matrix not equal to zero. There is a unique solutionb)   a  =      5    2    ,  b  =  2The determinant is zero, but the two lines are identical, There are infinitely many solutionc)   a  b  =  5  ∧  ¬  (  a  =      5    2    ∧  b  =  2  )The determinant is zero, but the two lines are parallel. There are no solutionsThe determinant   ⇒  (  5  −  a  b  )