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.