1. For each of the following, decide whether or not it is possible for a system
ID: 3169203 • Letter: 1
Question
1. For each of the following, decide whether or not it is possible for a system to satisfy the given description. If it is possible, give an augmented matrix (in row-echelon or reduced row-echelon form) that corresponds to such a system and prove that the corresponding system does in fact fulfill the requirements; if it is not possible, prove that it is not possible. (a) A system of 5 equations in 3 unknowns that has no solutions. (b) A system of 5 equations in 3 unknowns that has exactly 1 solution. (c) A system of 5 equations in 3 unknowns that has infinitely many solutions (d) A system of 5 equations in 3 unknowns that has exactly 2 solutions.Explanation / Answer
1.(a). It is possible for a system of 5 equations in 3 unknowns not to have a solution. Let the equations be x+y +z = 2; x+y-z= 4; 2x+y-z = 6; x+2y+3z=1 and 3x+3y+3z= 5. This system has no solution.
(b). It is possible for a system of 5 equations in 3 unknowns to have exactly 1 solution. Let the equations be x+y+z = 3; x-y+z = 1; x-y-z= -1; 2x+2y+2z = 6 and 3x-3y+3z =3. This system has a unique solution (x,y,z) = (1,1,1). The augmented matrix of the system is A =
1
1
1
3
1
-1
1
1
1
-1
-1
-1
2
2
2
6
3
-3
3
3
The RREF of A is
1
0
0
1
0
1
0
1
0
0
1
1
0
0
0
0
0
0
0
0
(c ). It is possible for a system of 5 equations in 3 unknowns to have infinitely many solutions. Let the equations be x+y+z = 3; x-y+z = 1; 2x+2y+2z=6; 3x-3y+3z=3 and 3x+3y+3z= 9. The augmented matrix of the system is A =
1
1
1
3
1
-1
1
1
2
2
2
6
3
-3
3
3
3
3
3
9
The RREF of A is
1
0
1
2
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
The solutionof this system is x+z= 2, y =1. For an arbitrary value t (say) of z, x will a value 2-t. Since t is arbitrary, there will be infinite solutions.
(d). It is not possible for a system of 5 equations in 3 unknowns to have exactly 2 solutions. Such a system of 5 linear equations in 3 variables may have no solution, one solution , or infinitely many solutions.
1
1
1
3
1
-1
1
1
1
-1
-1
-1
2
2
2
6
3
-3
3
3
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