Problem 58 - Precalculus with Limits

Question

In Exercises \(55-58\) , write the system of linear equations represented by the augmented matrix. Then use back. substitution to solve. (Use variables \(x, y,\) and \(z,\) if applicable.) $$\left[\begin{array}{rrrrr}{1} & {2} & {-2} & {\vdots} & {-1} \\ {0} & {1} & {1} & {\vdots} & {9} \\ {0} & {0} & {1} & {\vdots} & {-3}\end{array}\right]$$

Step-by-Step Solution

Answer

The solutions are \(x = -27\), \(y =12\), \(z =-3\).

Step 1: Decode the Augmented Matrix

Conversion of the Given Augmented matrix into a system of linear equations will result in: \( x + 2y - 2z = -1\), \( y + z = 9\), and \( z = -3\)

Step 2: Back Substitution

We will start by substituting \( z = -3 \) into the second equation, and then substitute \( z = -3 \) and \( y = 12 \) into equation 1. This gives us: \( y + (-3) = 9 \) simplifying to \( y = 12 \) and \( x + 2*(12) - 2*(-3) = -1 \) simplifying to \( x = -27 \)