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The augmented matrix is a powerful tool in solving systems of linear equations by using a compact and organized way of representing them. An augmented matrix consists of the coefficients of the variables in your equations lined up in columns, with a special column added – the augmented part – that contains the constants from each equation. This extra column is separated by a vertical line or a colon to remind us of its special role.

For example, the matrix \[\begin{matrix}1 & 8 & \vdots & 12 \0 & 1 & \vdots & 3 \\end{matrix}\]represents two equations:

• The first row gives the equation \(x + 8y = 12\).
• The second row shows the equation \(y = 3\).

They provide a clear and organized way to work through the system to find the solution systematically and efficiently. Understanding how to construct and interpret an augmented matrix is essential for solving linear systems effectively.

Back-substitution is a step-wise method used to solve linear systems, especially when working with matrices in row-echelon or reduced row-echelon form. It's like working backward from the last equation to the first to find the values of unknowns.

Once we have an easily solvable equation, we start by finding the value of the variable presented most simply (like having a coefficient of 1). In the example, the matrix is already simplified, showing \(y=3\).

With this value at hand, we replace or "substitute" it back into the previous equations to find other variables. For instance:
- Start with \(y=3\). You already know \(y\).'s value from the second row.- Substitute \(y=3\) into the first equation \(x + 8y = 12\) to find \(x\).

We solve \(x + 8\times3 = 12\), which simplifies to: \(x = 12 - 24\equiv x = -12\).

This step-by-step approach is straightforward but requires careful attention at each stage to ensure accuracy.

A system of linear equations consists of multiple linear equations that we aim to solve all together. These systems form the basis of many mathematical applications, from simple problems to complex real-world phenomena.

In the given example, the system comprised the two equations:

• \(x + 8y = 12\)
• \(y = 3\)

These lend themselves to finding the precise values of \(x\) and \(y\) that satisfy both conditions simultaneously. Solving such systems can tell us a lot about relationships between different variables in practical situations.
Each solution to a system of equations represents a point of intersection if interpreted graphically. In our example, solving the system gives us the coordinate \((-12, 3)\), which visually is where the lines \(x + 8y = 12\) and \(y=3\) cross in a two-dimensional plane.