Problem 59 - Precalculus with Limits

Question

In Exercises \(59-62,\) an augmented matrix that represents a system of linear equations (in variables \(x, y,\) and \(z,\) if applicable) has been reduced using Gauss.Jordan elimination. Write the solution represented by the augmented matrix. $$\left[\begin{array}{llll}{1} & {0} & {\vdots} & {3} \\ {0} & {1} & {\vdots} & {-4}\end{array}\right]$$

Step-by-Step Solution

Answer

The solution for the system of equations represented by the given augmented matrix is: \(x = 3\) and \(y = -4\).

Step 1: Understand Gauss-Jordan Elimination

Firstly, it is vital to understand Gauss-Jordan elimination. This method helps to solve a system of linear equations. By transforming the system's augmented matrix into reduced row-echelon form, one can straightforwardly read the solutions from it. The matrix in this exercise is already in this form.

Step 2: Interpret the Augmented Matrix

The given augmented matrix is: \[ \left[\begin{array}{rrr|r} 1 & 0 & \vdots & 3 \ 0 & 1 & \vdots & -4 \ \end{array}\right] \]This can be read as a system of linear equations where each row represents an equation and each column stands for a variable coefficient (except the last one, which represents the constant terms). The vertical bar separates the coefficients of variables from the constant terms. Here, the first row corresponds to \(1x + 0y = 3\), and the second row to \(0x + 1y = -4\).

Step 3: Write Out the Solution

With the matrix interpreted, the solution to the system can be written directly: \(x = 3\) and \(y = -4\). The values on the right side of the dotted line are the solutions for each corresponding variable.