Problem 99 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use Gauss-Jordan elimination to solve the system of equations. $$\left\\{\begin{array}{l} 2 x-y+7 z=-10 \\ 3 x+2 y-4 z=17 \\ 6 x-5 y+z=-20 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system is \(x = -1\), \(y = 3\), and \(z = 1\).

Step 1: Setup the Augmented Matrix

First, convert the given system of equations into an augmented matrix:\n \[\begin{bmatrix} 2 & -1 & 7 & -10\\ 3 & 2 & -4 & 17\\ 6 & -5 & 1 & -20 \end{bmatrix}\]

Step 2: Eliminate x from Second and Third Equation

Perform row operations to eliminate \(x\) from the second and third equation. Multiply the first row by 3/2 and subtract it from the second row. Also, multiply the first row by 3 and subtract from the third row.\n \[\begin{bmatrix} 2 & -1 & 7 & -10\\ 0 & 2.5 & -14.5 & 32\\ 0 & -2 & -20 & 10 \end{bmatrix}\]

Step 3: Eliminate \(y\) from third equation

Now, we eliminate \(y\) from the third equation. For that, multiply the second row by 4/5 and add to the third row.\n \[\begin{bmatrix} 2 & -1 & 7 & -10\\ 0 & 2.5 & -14.5 & 32\\ 0 & 0 & -6 & -6 \end{bmatrix}\]

Step 4: Convert to Reduced Row Echelon Form

To convert the matrix into its Reduced Row Echelon Form, divide the third row by -6, the second row by 2.5 and the first row by 2. Also, add the second row to first to eliminate \(y\) from first equation.\n \[\begin{bmatrix} 1 & 0 & 0 & -1\\ 0 & 1 & 0 & 3\\ 0 & 0 & 1 & 1 \end{bmatrix}\]

Step 5: Interpret the Solution

Looking at the final matrix, we can convert back into system of equations: \n \(x = -1\), \n \(y = 3\), and \n \(z = 1\).

Key Concepts

Augmented Matrix

An augmented matrix is a vital step in solving a system of equations using the Gauss-Jordan elimination method. It consists of the coefficient matrix of the system along with an extra column representing the constant terms from the equations. This combined form helps streamline the process of applying row operations and simplifies tracking changes.

Consider the original system of equations:

• \( 2x - y + 7z = -10 \)
• \( 3x + 2y - 4z = 17 \)
• \( 6x - 5y + z = -20 \)

We convert this system into an augmented matrix:\[\begin{bmatrix} 2 & -1 & 7 & -10\ 3 & 2 & -4 & 17\ 6 & -5 & 1 & -20 \end{bmatrix}\] The first three columns represent the coefficients of \(x\), \(y\), and \(z\), and the last column corresponds to the constants on the right side of the equations.

Row Operations

Row operations are the manipulations we perform on the rows of an augmented matrix to achieve a simpler form. They play a central role in transforming the matrix toward reduced row echelon form (RREF), where the solution becomes apparent. There are three types of row operations:

• Swapping two rows: Simply switch the position of any two rows.
• Multiplying a row by a nonzero scalar: This involves scaling all entries of a row by a constant value.
• Adding or subtracting a multiple of one row to another row: Use this to create zeros in strategic positions.

These operations help us eliminate variables column by column. In the given example, we use row operations to eliminate \(x\) from the second and third equations, and then eliminate \(y\) from the third equation. These steps are crucial for simplifying the matrix and making it more manageable for further calculations.

Reduced Row Echelon Form

When solving systems of equations using matrices, reaching the reduced row echelon form (RREF) is the ultimate goal. RREF is a special type of matrix where each leading entry in a row is 1, and all other entries in the column containing the leading entry are zeros. Moreover, leading entries proceed strictly downwards in subsequent rows.

In the example, we convert the following matrix: \[\begin{bmatrix} 2 & -1 & 7 & -10\ 0 & 2.5 & -14.5 & 32\ 0 & 0 & -6 & -6 \end{bmatrix}\]into the RREF:\[\begin{bmatrix} 1 & 0 & 0 & -1\ 0 & 1 & 0 & 3\ 0 & 0 & 1 & 1 \end{bmatrix}\]By dividing rows to have leading ones and performing additional operations to clear values in respective columns, we've organized the matrix beautifully. The final column now directly represents the solutions to the system.

System of Equations

A system of equations is a collection of multiple equations, usually with more than one unknown variable that we strive to solve simultaneously. Solutions for a system typically include values for each variable that satisfy all equations in the system at once.

The system given in this exercise consists of three equations in three variables \(x\), \(y\), and \(z\):

• \( 2x - y + 7z = -10 \)
• \( 3x + 2y - 4z = 17 \)
• \( 6x - 5y + z = -20 \)

Gauss-Jordan elimination helps simplify such systems by transforming them into a form where the solutions, if they exist, become clear and straightforward. In this case, the solution is finding the values \(x = -1\), \(y = 3\), and \(z = 1\), which satisfy all three equations.