Problem 60 - 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. $$\left[\begin{array}{llll}{1} & {0} & {\vdots} & {-6} \\ {0} & {1} & {\vdots} & {10}\end{array}\right]$$

Step-by-Step Solution

Answer

The solution to the system represented by the augmented matrix is \(x = -6\) and \(y = 10\).

Step 1: Understanding matrix

The matrix \[\left[\begin{array}{rrrrr} 1 && 0 && \vdots && -6 \ 0 && 1 && \vdots && 10 \ \end{array}\right]\] represents a system of equations, in which each horizontal line represents a unique equation. The first number in the line represents the coefficient of \(x\), the second one the coefficient of \(y\) and the last number on the right hand side of \(\vdots\) represents the result of the equation.

Step 2: Converting augmented matrix to system of equations

The first equation represented by the first line is: \(1x + 0y = -6\), which simplifies to \(x = -6\). The second equation represented by the second line is: \(0x + 1y = 10\), simplified it becomes \(y = 10\).

Step 3: Interpretation of the solution

The solutions of the system of equations derived from the augmented matrix are \(x = -6\) and \(y = 10\).