Proving Consistency in Linear Systems: The Case of Three Equations

[CaityAnn]

I have three systems of equations:
x+y+2z=a , x+z= b and 2x+y+3z=c
Show that in order for this system to have at least one solution, a,b,c must satisfy c=a+b.

Obviously I can add the equations a and b and get c. But I don't know how else to approach showing this. I think the points of x,y,z of c must satisfy both a,b and provide a solution set for both but I am not sure how to prove that. HELP PLEASE~!

 

[courtrigrad]

The matrix system is:

\begin{bmatrix}<br /> 1 &amp; 1 &amp; 2 &amp; a \\<br /> 1 &amp; 0 &amp; 1 &amp; b \\<br /> 2 &amp; 1 &amp; 3 &amp; c \\<br /> \end{bmatrix}

The system has at least one solution if the rank of the coefficient matrix equals the rank of the augmented matrix. Get the matrix to row-echleon form.