Consider the system of equations

Question

Consider the following system of equations

$x_1+2x_2 =k$

$3x_1+4x_2+x_3 =1$

$5x_1+6x_2+2x_3=2$

where $k$ is an undetermined constant. For what values of $k$, if any, is the system inconsistent?

Answer 1

Our goal is to construct an augmented matrix for these equations, simplify by performing some row operations, and to see if we can infer something about $k$. $$ \left[\begin{array}{rrr|r} 1 & 2 & 0 & k \\ 3 & 4 & 1 & 1 \\ 5 & 6 & 2 & 2 \end{array}\right] $$

As Tanner suggested, multiply the second row by 2 ($2R_2$):

$$ \left[\begin{array}{rrr|r} 1 & 2 & 0 & k \\ 6 & 8 & 2 & 2 \\ 5 & 6 & 2 & 2 \end{array}\right] $$

We now see that the last two columns of Row 2 reduce to 0 if we subtract Row 3 from it (i.e. $R_2-R_3$):

$$ \left[\begin{array}{rrr|r} 1 & 2 & 0 & k \\ 1 & 2 & 0 & 0 \\ 5 & 6 & 2 & 2 \end{array}\right] $$

And now the coefficient sides of $R_1$ and $R_2$ are identical, so we subtract $R_1-R_2$ to get $k$ on its own:

$$ \left[\begin{array}{rrr|r} 0 & 0 & 0 & k \\ 1 & 2 & 0 & 0 \\ 5 & 6 & 2 & 2 \end{array}\right] $$

We now see that if $k\neq0$, there are no possible solutions (note that if $k=0$ then we would have infinitely many solutions). The system is therefore inconsistent if $k\neq0$.

Answer 2

Hint:

Multiply the second equation by $2$ and subtract the third equation.

What must $k$ be to be consistent?

Answer 3

In the eq. $AX=B$ if $|A|=0$ but at least one of the Cramer's determinants $|A_1|, | ,A_2|,|A_3|$ is non-zero. $$|A|=\left |\begin{array}{ccc} 1 & 2 & 0 \\ 3 & 4 & 1 \\ 5 & 6 & 2\end{array} \right |=0$$ $$|A_1|=\left |\begin{array}{ccc} k & 2 & 0 \\ 2 & 4 & 1 \\ 2 & 6 & 2\end{array} \right |=2k-4 $$ $$|A_2|= \left| \begin{array}{ccc} 1 & k & 0\\ 3 & 2 & 1 \\5 & 2 & 2 \end{array} \right|=2-k$$ $$ \left| \begin{array}{ccc} 1 & 2 & k \\3 & 4 & 2 \\ 5 & 6 & 2 \end{array}\right|=4-2k$$

So If $k=2$ there are many solutions and for $k \ne 2$ there will be no solution and the system will be inconsistent.