Hello, could someone please help me out with the following questions?(adsbygoogle = window.adsbygoogle || []).push({});

Q. Determine the conditions on a, b, c so that the following systems are consistent and find any solutions.

a)

[tex]

\begin{array}{l}

x + 2y - 3z = a \\

3x - y + 2z = b \\

x - 5y + 8z = c \\

\end{array}

[/tex]

b)

[tex]

\begin{array}{l}

x + 2y + 4z = a \\

2x + 3y - z = b \\

3x + y + 2z = c \\

\end{array}

[/tex]

Firstly, the 'solutions' that I end up getting seem to either require me to introduce a parameter(for the first system) for one of the variables or the answer is in terms of a, b and c(for the seond systrem).

I am really unsure about how to do this question. For part a I wrote the corresponding augmented matrix and got down to what I think is it's reduced echelon form.

[tex]

\left( {\left| {\begin{array}{*{20}c}

1 & 2 & { - 3} \\

0 & { - 7} & {11} \\

0 & 0 & 0 \\

\end{array}} \right|\begin{array}{*{20}c}

a \\

{b - 3a} \\

{c + 2a - b} \\

\end{array}} \right)

[/tex]

So the condition need for the system to be consistent that c+2a-b = 0 right? I'm not sure what else needs to be said to determine the conditions for a, b and c so that the system is consistent. Also, when I solved the equation I let z = s where s is a real number so that I got:

[tex]

\left( {x,y,z} \right) = \left( {\frac{{5a + 2c + s}}{7},\frac{{11s + a - c}}{7},s} \right)

[/tex]

For part b I went further so as to reduce the augmented matrix to row reduced echelon form:

[tex]

\left( {\left| {\begin{array}{*{20}c}

1 & 0 & 0 \\

0 & 1 & 0 \\

0 & 0 & 1 \\

\end{array}} \right|\begin{array}{*{20}c}

{\frac{{140b - 259a - 14c}}{{35}}} \\

{\frac{{133a - 80b + 9c}}{{35}}} \\

{\frac{{7a - 5b + c}}{{35}}} \\

\end{array}} \right)

[/tex]

Now the 'problem' with this is that none of the rows in the coefficient matrix are zero. So no matter what I try to do with the overall augmented matrix, it seems that the system is always consistent. Can this possibly be right? Also, from the augmented matrix the 'solution' to the system would be:

[tex]

\left( {x,y,z} \right) = \left( {\frac{{140b - 259a - 14c}}{{35}},\frac{{133a - 80b + 9c}}{{35}},\frac{{7a - 5b + c}}{{35}}} \right)

[/tex]

Now that just looks way too complicated to be correct. Can someone please help me out with this question? Any help would be great thanks.

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# Systems of equations - consistency

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