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. 
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        This assignment is of the kind  " Nothing can be easier ".



<pre>
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.
</pre>

Solved, &nbsp;with explanations.


Hope, &nbsp;everything is clear to you.


If you still have questions, &nbsp;do not hesitate to ask.



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Let's imagine for a minute, &nbsp;that the original system consists of two equations &nbsp;(1), &nbsp;(2) 
PLUS &nbsp;60 &nbsp;other equations, &nbsp;identical to equation &nbsp;(3).


Then the analysis is the same: &nbsp;this system of &nbsp;62 &nbsp;equations is overdetermined, 
but it is &nbsp;EQUIVALENT &nbsp;to the consistent system of two equations &nbsp;(1) &nbsp;and &nbsp;(2).


Again, &nbsp;it has the &nbsp;UNIQUE &nbsp;solution &nbsp;x= 5, &nbsp;y= -7.



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Hope, &nbsp;in the future you will not afraid such problems.