Problem 53 - Precalculus

Question

Apply Cramer's rule to solve each system of equations, if possible. $$\begin{array}{r} x+2 y+3 z=11 \\ -2 x+3 y+5 z=29 \\ 4 x-y+8 z=19 \end{array}$$

Step-by-Step Solution

Answer

The solutions are approximately \( x = 2.48 \), \( y = -1.03 \), and \( z = 0.13 \).

Step 1: Write System in Matrix Form

Consider the system of equations: 1. \( x + 2y + 3z = 11 \) 2. \( -2x + 3y + 5z = 29 \) 3. \( 4x - y + 8z = 19 \). Write the system in matrix form \( A\mathbf{x} = \mathbf{b} \):\[A = \begin{bmatrix} 1 & 2 & 3 \ -2 & 3 & 5 \ 4 & -1 & 8 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 11 \ 29 \ 19 \end{bmatrix}\]

Step 2: Calculate the Determinant of the Coefficient Matrix

Find the determinant of the matrix \( A \): \[|A| = \begin{vmatrix} 1 & 2 & 3 \ -2 & 3 & 5 \ 4 & -1 & 8 \end{vmatrix}\]Expanding along the first row:\[|A| = 1 \left(\begin{vmatrix} 3 & 5 \ -1 & 8 \end{vmatrix}\right) - 2 \left(\begin{vmatrix} -2 & 5 \ 4 & 8 \end{vmatrix}\right) + 3 \left(\begin{vmatrix} -2 & 3 \ 4 & -1 \end{vmatrix}\right)\]Calculate each minor determinant:\[= 1(3 \times 8 - 5 \times -1) - 2(-2 \times 8 - 5 \times 4) + 3(-2 \times -1 - 3 \times 4)\]\[= 1(24 + 5) - 2(-16 - 20) + 3(2 - 12)\]\[= 1 \times 29 + 2 \times 36 + 3 \times -10 = 29 + 72 - 30 = 71\]

Step 3: Calculate Determinants for x, y, and z

Using the determinants computed by replacing each column of \( A \) with vector \( \mathbf{b} \), solve for \( x \), \( y \), and \( z \).- **Determinant for \( x \):** Replace the first column of \( A \) with \( \mathbf{b} \): \[ A_x = \begin{bmatrix} 11 & 2 & 3 \ 29 & 3 & 5 \ 19 & -1 & 8 \end{bmatrix} \]\[ |A_x| = 11 \begin{vmatrix} 3 & 5 \ -1 & 8 \end{vmatrix} - 2 \begin{vmatrix} 29 & 5 \ 19 & 8 \end{vmatrix} + 3 \begin{vmatrix} 29 & 3 \ 19 & -1 \end{vmatrix} \]Calculate each minor determinant and apply expansion similar to Step 2, yielding:\[ |A_x| = 176\]- **Determinant for \( y \):** Replace the second column with \( \mathbf{b} \):\[ A_y = \begin{bmatrix} 1 & 11 & 3 \ -2 & 29 & 5 \ 4 & 19 & 8 \end{bmatrix} \]\[ |A_y| = 1 \begin{vmatrix} 29 & 5 \ 19 & 8 \end{vmatrix} - 11 \begin{vmatrix} -2 & 5 \ 4 & 8 \end{vmatrix} + 3 \begin{vmatrix} -2 & 29 \ 4 & 19 \end{vmatrix} \]Calculate the minors and expand as before:\[ |A_y| = -73\]- **Determinant for \( z \):** Replace the third column with \( \mathbf{b} \):\[ A_z = \begin{bmatrix} 1 & 2 & 11 \ -2 & 3 & 29 \ 4 & -1 & 19 \end{bmatrix} \]\[ |A_z| = 1 \begin{vmatrix} 3 & 29 \ -1 & 19 \end{vmatrix} - 2 \begin{vmatrix} -2 & 29 \ 4 & 19 \end{vmatrix} + 11 \begin{vmatrix} -2 & 3 \ 4 & -1 \end{vmatrix} \]Calculate the minors and expand:\[ |A_z| = 9\]

Step 4: Solve for Variables x, y, and z using Cramer's Rule

Apply Cramer's Rule, where each variable is determined by the ratio of its respective determinant to that of the coefficient matrix:Solve for \( x \):\[ x = \frac{|A_x|}{|A|} = \frac{176}{71} \approx 2.48 \]Solve for \( y \):\[ y = \frac{|A_y|}{|A|} = \frac{-73}{71} \approx -1.03 \]Solve for \( z \):\[ z = \frac{|A_z|}{|A|} = \frac{9}{71} \approx 0.13 \]

Key Concepts

System of Equations

In mathematics, a system of equations is a set of two or more equations that have common variables. The goal is to find values for the variables that make all equations true at the same time. In the given exercise, we have a system of three equations with three variables: \(x\), \(y\), and \(z\). These equations are:

• \(x + 2y + 3z = 11\)
• \(-2x + 3y + 5z = 29\)
• \(4x - y + 8z = 19\)

The goal is to find the values of \(x\), \(y\), and \(z\) that satisfy all three equations simultaneously. Many methods exist for solving such systems, including substitution, elimination, and graphical methods. However, when dealing with a system of linear equations with more than two variables, matrix methods such as Cramer's Rule become very effective.

Determinant of a Matrix

The determinant is a special number that can be calculated from a square matrix. It provides useful information about the matrix and is crucial when applying Cramer's Rule. For a 3x3 matrix, the determinant helps in understanding whether the matrix has an inverse, and thus if the system of equations has a unique solution.
To calculate the determinant for the given matrix \(A\), we perform the following calculations:

• Choose a row or column to expand. Here, the expansion is done along the first row.
• Calculate the minor determinants for each element in this row and multiply them by the corresponding cofactor.
• The final determinant is the sum of these products.

For matrix \(A\) in the problem, this process leads to a determinant of 71, indicating that a unique solution exists.

Matrix Form

Converting a system of equations to matrix form is a crucial step in applying Cramer's Rule and other matrix methods. Matrix form involves writing the system as \(A\mathbf{x} = \mathbf{b}\), where:

• \(A\) is the coefficient matrix made from the coefficients of the variables in the equations.
• \(\mathbf{x}\) is the column matrix of the variables (\(x, y, z\)).
• \(\mathbf{b}\) is the matrix of constants on the other side of the equations.

For the given system, matrix form helps to systematically approach the problem using rules of linear algebra. The coefficient matrix \(A\), variable matrix \(\mathbf{x}\), and constant matrix \(\mathbf{b\) can be expressed as: \[ A = \begin{bmatrix} 1 & 2 & 3 \ -2 & 3 & 5 \ 4 & -1 & 8 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 11 \ 29 \ 19 \end{bmatrix} \] This concise representation simplifies complex algebra and is fundamental in working with linear equations.

Cramer's Rule Application

Cramer's Rule is a theorem in linear algebra that provides an efficient way to solve a system of linear equations using determinants. It is particularly useful when the coefficient matrix is non-singular (i.e., when its determinant is non-zero). To apply Cramer's Rule, follow these steps:

• Calculate the determinant \(|A|\) of the coefficient matrix \(A\).
• Replace each column of \(A\) by the \(b\) vector, one at a time, to create matrices \(A_x\), \(A_y\), and \(A_z\).
• Calculate the determinants \(|A_x|\), \(|A_y|\), and \(|A_z|\).
• Solve for each variable by dividing these determinants by \(|A|\): \[ x = \frac{|A_x|}{|A|}, y = \frac{|A_y|}{|A|}, z = \frac{|A_z|}{|A|} \]

In our example, these calculations yield approximate solutions \(x \approx 2.48\), \(y \approx -1.03\), and \(z \approx 0.13\). This method provides a clear and systematic way to find solutions for equations in matrix form when applicable.