Question

Suppose the system below is consistent for all possible values of f and g. What can you say about the coefficients c and d? Justify your answer. x1 + 3x2 = f cx1 + dx2 = g

Answer 1

Answer:

The answer is "$d \neq 3c,$f and g are arbitrary".

Step-by-step explanation:

The matrix of the device is increased  

$\left[\begin{array}{ccc}1&3&f\\ c&d&g\\ \end{array}\right]$

Reduce the echelon row matrix  

$\left[\begin{array}{ccc}1&3&f\\ c&d&g\\ \end{array}\right] \\\\R_1 \leftrightarrow R_2 \\\\\frac{R_2 -1}{C R_1 \to R_2} \sim \left[\begin{array}{ccc} c&d&g\\ 0 & \frac{3c-d}{c}& \frac{cf-g}{c}\\ \end{array}\right]$

Therefore, if 3c$\neq$ 0 is d $\neq$  3c, the device is valid.  Therefore d $\neq$ are arbitrary 3c, g and f.