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Question 1202811: Given the following augmented matrix in row-reduced form, assume that it is equivalent to an augmented matrix corresponding to a system of linear equations.
[1 0 | 5]
[0 1 | −7]
[0 0 | 0]
(a.) Determine the number of equations and number of variables in the corresponding system of linear equations.
Number of Equations: 3
Number of Variables: 2
(b.) Determine whether the system is underdetermined or overdetermined.
Underdetermined
Overdetermined
.
(c.) Find the solution(s) to the system, if it exists. State the solution as a point, using the parameter(s) s and t if needed. If the system is inconsistent, then enter NO SOLUTION.
Part c is where I am stuck. I got all parts but part c correct, I'm not sure of an easy way or a full way of knowing how to figure this out. I got to close to the solution for x, but I'm still lost.
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
The corresponding system of equations is
(1) 1x+0y = 5
(2) 0x+1y = -7
(3) 0x+0y = 0
part a: That is clearly 3 equations in 2 variables.
part b: The formal mathematical definition of overdetermined is that there are more equations than there are variables. So, despite that fact that the third equation provides no useful information that can contradict the solution obtained from the first two equations, this system is overdetermined.
part c: The first two equations directly from the matrix give the single solution to the system: (x,y) = (5,-7)
Answer by ikleyn(52787)  (Show Source):
You can put this solution on YOUR website! Given the following augmented matrix in row-reduced form,
assume that it is equivalent to an augmented matrix corresponding to a system of linear equations.
[1 0 | 5]
[0 1 | −7]
[0 0 | 0]
(a.) Determine the number of equations and number of variables in the corresponding system of linear equations.
Number of Equations: 3
Number of Variables: 2
(b.) Determine whether the system is underdetermined or overdetermined.
Underdetermined
Overdetermined
.
(c.) Find the solution(s) to the system, if it exists. State the solution as a point, using the parameter(s) s and t if needed.
If the system is inconsistent, then enter NO SOLUTION.
Part c is where I am stuck. I got all parts but part c correct, I'm not sure of an easy way or a full way of knowing
how to figure this out. I got to close to the solution for x, but I'm still lost.
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This assignment is of the kind " Nothing can be easier ".
The given augmented matrix represents the system of equations
1x + 0y = 5 (1)
0x + 1y = -7 (2)
0x + 0y = 0 (3)
It formally has two variables / unknowns x and y and three equations (1), (2) and (3).
Formally, this system is overdetermined (2 unknowns and 3 equations).
But in reality, the third equation is valid for any values of x and y,
so, in a meaningful context, it does not carry any information.
Therefore, we can remove it from the system and consider further TWO equations (1) and (2) only.
Then the system becomes
x + 0y = 5 (4)
0x + y = -7 (5)
and can be written even SIMPLER
x = 5 (6)
y = -7 (7)
These equalities, (6) and (7) represent the UNIQUE solution for the given system.
Thus the original system of three equations (1), (2) and (3) is formally overdetermined;
but in reality, it is equivalent to the CONSISTENT system of two equations (4), (5),
which has the UNIQUE solution x= 5, y= -7.
Solved, with explanations.
Hope, everything is clear to you.
If you still have questions, do not hesitate to ask.
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Let's imagine for a minute, that the original system consists of two equations (1), (2)
PLUS 60 other equations, identical to equation (3).
Then the analysis is the same: this system of 62 equations is overdetermined,
but it is EQUIVALENT to the consistent system of two equations (1) and (2).
Again, it has the UNIQUE solution x= 5, y= -7.
.................
Hope, in the future you will not afraid such problems.
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