### Video Transcript

From the augmented matrix two,
zero, negative nine, five, zero, four, negative nine, five, negative four, negative
nine, zero, zero, find the system of equations.

We begin by assuming that the
variables corresponding to the first, second, and third columns should be labeled as
𝑥, 𝑦, and 𝑧, respectively. This means that we need to populate
the missing entries in the following system of three equations in three
unknowns. The first column of our augmented
matrix corresponds to the 𝑥-coefficients. These are two, zero, and negative
four. The second column corresponds to
the 𝑦-coefficients: zero, four, and negative nine. The third column, negative nine,
negative nine, and zero are the 𝑧-coefficients. Finally, the elements in the
right-hand column of our matrix correspond to the entries on the right-hand side of
our equations. These are five, five, and zero.

We can simplify these equations by
ignoring any term that has a coefficient of zero. As adding negative nine is the same
as subtracting nine. The first equation can be rewritten
as two 𝑥 minus nine 𝑧 is equal to five. In the same way, the second
equation becomes four 𝑦 minus nine 𝑧 equals five and the third equation becomes
negative four 𝑥 minus nine 𝑦 equals zero. The augmented matrix two, zero,
negative nine, five, zero, four, negative nine, five, negative four, negative nine,
zero, zero corresponds to the system of equations two 𝑥 minus nine 𝑧 equals five,
four 𝑦 minus nine 𝑧 equals five, and negative four 𝑥 minus nine 𝑦 equals
zero.