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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.$$\begin{aligned} x_{1}+3 x_{2} &=f \\ c x_{1}+d x_{2} &=g \end{aligned}$$

$d \neq 3 c$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 1

Systems of Linear Equations

Introduction to Matrices

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So in this problem, we're told that this system is consistent for all values of F G and were asked what we can deduce about the values of C and D. And so what we're going to do is look at this sort of left hand matrix here and take the determinant. And we know that for this to be consistent, the determinant can't be zero. So remember that we do the determinant by doing kind of this curly fish pattern. So it will be one times d minus three times, see, and that we don't want to be equal to zero in order for this to be consistent, which means that DE cannot equal three c.

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