91Ó°ÊÓ

When faced with a system of linear equations, like the pair \(2x - 3y = 2\) and \(6x - 9y = 3\), we're essentially looking at multiple linear equations that we are aiming to solve together. Each equation represents a straight line, and the solutions to the system are the points where the lines intersect.

In our case, the two equations have the same ratio for the coefficients of \(x\) and \(y\), which suggests that the lines may either be parallel (no intersection) or coincident (infinite intersections). To determine which scenario they present, we typically manipulate the equations in a way that allows us to either isolate one variable in terms of the others or identify if there's an inconsistency or dependency.

Elementary row operations are the bread and butter of solving systems of equations with matrices. They involve three types of moves we can perform:

• Swapping two rows
• Multiplying a row by a non-zero constant
• Adding or subtracting multiples of one row to another

These operations won't change the solution set of the system but will help to simplify the matrix into a form where the solutions can be read off directly.

In the given problem, dividing the first row by 2 and then using a multiple of the first row to eliminate the leading coefficient in the second row are excellent examples of elementary row operations. They tactfully transform the matrix into a simpler form that will ultimately lead us to the solution.

To get to the answers of our system, we aim for the Reduced Row Echelon Form (RREF). This special matrix format has certain characteristics:

• Each leading entry in a row is 1 (also known as a pivot).
• Each pivot is to the right of the pivots in the above row.
• All entries above and below each pivot are zeros.
• If there are rows of all zeros, they are at the bottom.

The Gauss-Jordan elimination method is a process we use to get to RREF, and once we're there, each equation represented by the matrix points directly to a solution.

However, in some cases, like the one in the exercise, reaching RREF reveals that we can't find a unique solution for each variable, indicating either a dependent system or a system with no solution at all.

An augmented matrix is a powerful tool to represent our system of linear equations compactly as a single matrix, which includes both the coefficients of the variables and the constants from the right-hand sides of the equations. Here's how it looks for our initial system:\[\left(\begin{array}{ccc}2 & -3 & 2 \6 & -9 & 3\end{array}\right)\]

This visual simplification allows us to apply elementary row operations across the entire matrix and focus on transforming it into RREF. The final step—interpreting the RREF—can effortlessly tell us the nature of the solutions. In some situations, as we've seen, the augmented matrix helps to highlight that some equations in the system are actually multiples of each other and therefore don't contribute additional information, revealing dependencies between the variables.