Problem 62 - 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. $$\left[\begin{array}{ccccc}{1} & {0} & {0} & {\vdots} & {5} \\ {0} & {1} & {0} & {\vdots} & {-3} \\ {0} & {0} & {1} & {\vdots} & {0}\end{array}\right]$$

Step-by-Step Solution

Answer

The system of equations represented by the matrix is \(x = 5\), \(y = -3\), and \(z = 0\). Therefore, the solution to the system is the ordered triple (5, -3, 0).

Step 1: Determine the matrix form

The matrix given has been reduced using Gauss-Jordan elimination to its simplest form which is a reduced row echelon form (RREF). In this form, each leading coefficient, appearing to the left of the other coefficients in its row, is 1, and the columns containing leading coefficients have zeros in all of their other entries.

Step 2: Convert the matrix into system of linear equations

To convert back into the system of equations, treat each line of the matrix as an equation. The columns represent the coefficients of variables x, y and z respectively and the last column represents the constant on the right side of the equation. This results in the system of equations:\[x = 5\]\[y = -3\]\[z = 0\]

Step 3: Write the solution set

The given system of linear equations is already solved and each equation represents the solution for its variable. Therefore, the solution can be expressed as an ordered triple that represents the values of the variables x, y, and z respectively.