Problem 44 - A Graphical Approach to Precalculus with Limits

Question

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} x+3 y-6 z &=7 \\ 2 x-y+2 z &=0 \\ x+y+2 z &=-1 \end{aligned}$$

Step-by-Step Solution

Answer

The solution is \(x = 1\), \(y = 0\), \(z = -1\).

Step 1: Write the Augmented Matrix

First, we will write the augmented matrix for the given system of equations. The system is: \(x+3y-6z=7\), \(2x-y+2z=0\), and \(x+y+2z=-1\). The augmented matrix representation is:\[\begin{bmatrix} 1 & 3 & -6 & | & 7 \2 & -1 & 2 & | & 0 \1 & 1 & 2 & | & -1 \end{bmatrix}\]

Step 2: Perform Row Operations to Obtain a Triangular Form

We need to transform the matrix to make the first column below the first row contain zeros. Start by subtracting Row 1 from Row 3 to replace Row 3. Then, subtract 2 times Row 1 from Row 2 to replace Row 2.\[\begin{bmatrix}1 & 3 & -6 & | & 7 \0 & -7 & 14 & | & -14 \0 & -2 & 8 & | & -8\end{bmatrix}\]

Step 3: Simplify the Matrix to Further Isolate Variables

To make the leading coefficient in Row 2 equal to 1, divide Row 2 by -7, and Row 3 by -2.\[\begin{bmatrix} 1 & 3 & -6 & | & 7 \0 & 1 & -2 & | & 2 \0 & 1 & -4 & | & 4\end{bmatrix}\]

Step 4: Eliminate Variables in Lower Rows

Subtract Row 2 from Row 3 to get:\[\begin{bmatrix} 1 & 3 & -6 & | & 7 \0 & 1 & -2 & | & 2 \0 & 0 & -2 & | & 2\end{bmatrix}\]

Step 5: Solve for the Variables

From the last row of the matrix, we have \(-2z = 2\), so \(z = -1\).Substitute \(z = -1\) in Row 2: \(y - 2(-1) = 2\) which gives \( y - (-2) = 2\), yielding \(y = 0\).Finally, substitute \(y = 0\) and \(z = -1\) in Row 1: \( x + 3(0) - 6(-1) = 7\) which simplifies to \( x + 0 + 6 = 7\), leading to \(x = 1\).

Step 6: Verify the Solution

Substitute \(x = 1\), \(y = 0\), and \(z = -1\) back into the original equations to ensure they satisfy all equations:1. \(1 + 3(0) - 6(-1) = 7\) - This is true.2. \(2(1) - (0) + 2(-1) = 0\) - This is true.3. \(1 + 0 + 2(-1) = -1\) - This is true. Thus, the solution is verified.

Key Concepts

Row Operations

Row operations are fundamental tools used in linear algebra to manipulate matrices and solve systems of equations. They allow us to transform matrices into simpler forms, making it easier to find solutions. There are three main types of row operations:

• Swapping two rows.
• Multiplying a row by a non-zero scalar.
• Adding or subtracting a multiple of one row to another row.

These operations are essential because they do not change the solutions of the system of equations represented by the matrix. In the given exercise, row operations help in systematically reducing the augmented matrix to a form where the solutions to the variables become more apparent.

System of Equations

A system of equations is a collection of multiple equations that share a set of variables. In this exercise, the system is comprised of three equations with three unknowns, represented as:

• \(x + 3y - 6z = 7\)
• \(2x - y + 2z = 0\)
• \(x + y + 2z = -1\)

The goal is to find values for the variables \(x\), \(y\), and \(z\) that simultaneously satisfy all of these equations. By turning the system into an augmented matrix and using row operations, we simplify the process of finding these values in an organized manner. This method avoids guesswork and provides a clear path to the solution.

Gauss-Jordan Elimination

Gauss-Jordan elimination is an algorithm used to solve systems of linear equations and find the inverse of matrices. It extends the Gaussian elimination method by bringing the matrix into reduced row-echelon form.
To perform Gauss-Jordan elimination, one transforms the augmented matrix by applying row operations until it attains the reduced form where:

• Each leading entry (leftmost non-zero entry) of a row is 1.
• Each leading 1 is the only non-zero entry in its column.

In this exercise, Gauss-Jordan elimination helps isolate each variable one at a time, making the matrix diagonal (or nearly diagonal) so that the solution can be read directly from the matrix. This step-by-step simplification significantly reduces computational complexity and enhances understanding.

Triangular Form

Triangular form, also known as upper triangular form, is a simplified matrix form that helps solve systems of equations efficiently. A matrix is in triangular form when all the elements below the main diagonal are zero. In an upper triangular form, we focus on positioning the zeros in such a manner that the matrix resembles a triangle.
For the given system of equations, transforming the augmented matrix into triangular form was achieved by making strategic row operations. The goal was to produce zero entries below the main diagonal, hence simplifying each equation step-by-step to isolate the variables effectively.
Triangular form serves as an intermediate step before reaching the reduced row-echelon form in the context of Gauss-Jordan elimination, facilitating an organized approach to back-solving the equations for the values of \(x\), \(y\), and \(z\).