Question

find an equation involving g, h, and k that makes this augmented matrix correspond to a consistent system:

Answer 1

The equation involving g, h, and k  is  mathematically given as

0=b ( "b" is equal to zero)

What is the equation involving g, h, and k that makes this augmented matrix correspond to a consistent system?

Generally, the equation for is  mathematically given as

$\begin{aligned}&{\left[\begin{array}{cccc}1 & -4 & 7 & g \\0 & 3 & -5 & h \\-2 & 5 & -9 & k\end{array}\right]} \\\\&\sim\left[\begin{array}{cccc}1 & -4 & 7 & g \\0 & 3 & -5 & h \\-2 & 5 & -9 & k\end{array}\right] \\\\&\sim\left[\begin{array}{cccc}1 & -4 & 7 & g \\0 & 3 & -5 & h \\0 & -3 & 5 & 2 g+k\\\\\end{array}\right] \quad R_{3}+2 R_{1} \\\\&\sim\left[\begin{array}{cccc}1 & -4 & 7 & g \\0 & 3 & -5 & h \\0 & 0 & 0 & 2 g+k+h\end{array}\right] \quad R_{3}+R_{2}\end{aligned}$

In conclusion,  Let "b" symbolize the integer 2g+k+h The final equation that is represented by the augmented matrix that was just given is thus 0=b.

Because this equation can only be satisfied if and only if the variable "b" is equal to zero, the first system can only have a solution if and only if the combination of g, k, and h equals 0.

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