for 10-12 you will solve the following system of equations: 2x+y+z=-2 2x-y+3z=6 3x-5y+4z=7 10) what is the solution for x? a)2 b)-3 c)infinitely many solutions d)no solution 11) what is the solution for y? a)2 b)0 c)inifinitely many solution d)no solution 12) what is the solution for z? a)4 b)-8 c)infinitely many solutions d)no solutions
The given system of equations is 2x+y+z=-2…(1), 2x-y+3z=6 …(2) and3x-5y+4z=7…(3).
The augmented matrix of this linear system is A (say) =
|
2 |
1 |
1 |
-2 |
|
2 |
-1 |
3 |
6 |
|
3 |
-5 |
4 |
7 |
To solve the given system of equations, we will reduce A to its RREF as under:
Multiply the 1st row by ½
Add -2 times the 1st row to the 2nd row
Add -3 times the 1st row to the 3rd row
Multiply the 2nd row by -1/2
Add 13/2 times the 2nd row to the 3rd row
Multiply the 3rd row by -1/4
Add 1 times the 3rd row to the 2nd row
Add -1/2 times the 3rd row to the 1st row
Add -1/2 times the 2nd row to the 1st row
Then the RREF of A is
|
1 |
0 |
0 |
-3 |
|
0 |
1 |
0 |
0 |
|
0 |
0 |
1 |
4 |
This implies that the given system of equations has a unique solution x = -3, y = 0 and z = 4.
10) b
11) b
12) a
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