Question Video: Finding a System of Equations from a Given Augmented Matrix Mathematics

From the augmented matrix (2, 0, −9, 5, 0, 4, −9, 5, −4, −9, 0, 0), find the system of equations.

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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.

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