Question: Examine the consistency of the following system of equations:
2x - 5y +7z = 6
3x - 8y + 11z = 11
x - 3y +4z = 3
A linear system of equations is consistent if each of the equations have a common solution. So, we attempt to find a common solution $\displaystyle {x,y,z} $ that satisfies each equation in the following linear system:
$\displaystyle 2x - 5y +7z = 6 $
$\displaystyle 3x - 8y + 11z = 11 $
$\displaystyle x - 3y +4z = 3 $.
You can use substitution, linear combination, Gaussian elimination or whichever is your preferred method for solving a linear system of equations.
Eventually, you will either get some solution {x,y,z} that satisfies the system of equations, or you will observe some ridiculous contradiction (i.e., 2=8) which is your cue that you have inconsistent equations of a linear system. Be sure to check your work. Holla back if you are still stuck.
-Andy
You can write this system in matrix form:
$\displaystyle A\mathbf{x} = \mathbf{b}$
with
$\displaystyle A = \left[\begin{matrix} 2 & -5 & 7\\ 3 & -8 & 11\\ 1 & -3 & 4 \end{matrix}\right]$.
Now find $\displaystyle |A|$. If it is nonzero, there will be a solution and the system will be consistent.
Well...
$\displaystyle |A| = 2\left|\begin{matrix}-8 & 11\\ -3 & 4 \end{matrix} \right| + 5\left| \begin{matrix} 3 & 11\\ 1 & 4\end{matrix}\right| + 7 \left|\begin{matrix}3 & -8\\ 1 & -3 \end{matrix}\right|$
$\displaystyle = 2(-8 \cdot 4 - 11 \cdot -3) + 5(3 \cdot 4 - 11 \cdot 1) + 7[3\cdot -3 - (-8)\cdot 1]$
$\displaystyle = 2\cdot 1 + 5 \cdot 1 + 7\cdot -1$
$\displaystyle = 2 + 5 - 7$
$\displaystyle = 0$.
So I would agree with your statement that the system is inconsistent.