Problem 62 - Intermediate Algebra

Question

Use Cramer's rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. $$ \left\\{\begin{array}{l} 2 x-y+4 z+2=0 \\ 5 x+8 y+7 z=-8 \\ x+3 y+z+3=0 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has solutions: \( x = \frac{9}{4} \), \( y = \frac{2}{7} \), \( z = \frac{13}{7} \).

Step 1: Write the system in standard form

First, rewrite the system of equations in standard form by moving all constants to the right side. The system becomes:\[\begin{align*}2x - y + 4z &= -2 \5x + 8y + 7z &= -8 \x + 3y + z &= -3\end{align*}\]

Step 2: Form the coefficient matrix A

The coefficient matrix \( A \) is formed from the coefficients of the variables \( x, y, z \), resulting in:\[A = \begin{bmatrix}2 & -1 & 4 \5 & 8 & 7 \1 & 3 & 1\end{bmatrix}\]

Step 3: Calculate determinant of matrix A

Calculate the determinant of matrix \( A \) using the formula for the determinant of a 3x3 matrix:\[\det(A) = 2(8 \times 1 - 7 \times 3) - (-1)(5 \times 1 - 7 \times 1) + 4(5 \times 3 - 8 \times 1)\]After simplifying, we find:\[\det(A) = -56\]

Step 4: Form the matrices A_x, A_y, A_z

Replace each column of \( A \) by the constant matrix \( \mathbf{b} = \begin{bmatrix} -2 \ -8 \ -3 \end{bmatrix} \) to form the matrices:- \( A_x = \begin{bmatrix} -2 & -1 & 4 \ -8 & 8 & 7 \ -3 & 3 & 1 \end{bmatrix} \)- \( A_y = \begin{bmatrix} 2 & -2 & 4 \ 5 & -8 & 7 \ 1 & -3 & 1 \end{bmatrix} \)- \( A_z = \begin{bmatrix} 2 & -1 & -2 \ 5 & 8 & -8 \ 1 & 3 & -3 \end{bmatrix} \)

Step 5: Calculate determinants for A_x, A_y, A_z

Calculate the determinants:\[\det(A_x) = (-2)(8) - (-1)(-8) + 4(5 \times 3 - 7 \times 1) = -126\]\[\det(A_y) = 2(-8) - (-2)(7) + 4(5 \times 1 - 1 \times 1) = -16\]\[\det(A_z) = 2(8 \times 1 - -8 \times 3) - (-1)(5 \times 1 - (-8)\times 1) = -104\]

Step 6: Use Cramer's Rule to find solutions

With \( \det(A) eq 0 \), solutions are given by Cramer's Rule:\[x = \frac{\det(A_x)}{\det(A)} = \frac{-126}{-56} = \frac{63}{28}\]\[y = \frac{\det(A_y)}{\det(A)} = \frac{-16}{-56} = \frac{4}{14}\]\[z = \frac{\det(A_z)}{\det(A)} = \frac{-104}{-56} = \frac{52}{28}\]

Step 7: Simplify the solutions

Simplify the fractions obtained:\[x = \frac{63}{28} = \frac{9}{4}, \quad y = \frac{4}{14} = \frac{2}{7}, \quad z = \frac{52}{28} = \frac{13}{7}\]

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special value that can be calculated from its elements. It plays a crucial role in linear algebra, particularly when solving systems of linear equations using Cramer's Rule. For a 3x3 matrix like the one in the given exercise, the determinant can be found using the formula: \[\det(A) = a(ei-fh) - b(di-fg) + c(dh-eg)\]where each term corresponds to specific elements of the matrix. Calculating the determinant helps us determine properties of the matrix, such as whether it is invertible.
An invertible matrix has a non-zero determinant, meaning it has a unique solution when used in linear systems. Conversely, a zero determinant indicates that the matrix is either dependent or inconsistent, leading to no solutions or infinitely many solutions.

System of Linear Equations

A system of linear equations consists of multiple linear equations, all involving the same set of variables. The aim is to find the values of these variables that satisfy all the equations simultaneously.
Consider:

• Two variables - Typically solved using substitution or elimination methods.
• Three or more variables - Systems like these often use matrices and determinants for solutions.

In our example, the system involves three equations with three variables \(x\), \(y\), and \(z\). In mathematical notation, it's expressed as:\[\begin{align*}2x - y + 4z &= -2 \5x + 8y + 7z &= -8 \x + 3y + z &= -3\end{align*}\]The goal is to find a set of \(x\), \(y\), and \(z\) values that satisfy these equations simultaneously, using methods like Cramer's Rule where applicable.

Steps to Solve Equations

Solving systems of equations using Cramer's Rule involves several sequential steps:1. **Writing in Standard Form**:
First, ensure each equation in the system is in standard form, which means all variables and coefficients are on one side, and constants are on the other. For instance, after adjustment: \[ \begin{align*} 2x - y + 4z &= -2 \ 5x + 8y + 7z &= -8 \ x + 3y + z &= -3 \end{align*} \]2. **Forming the Coefficient Matrix**:
Create a matrix \(A\) using the coefficients of \(x\), \(y\), and \(z\). The matrix looks like this: \[ A = \begin{bmatrix} 2 & -1 & 4 \ 5 & 8 & 7 \ 1 & 3 & 1 \end{bmatrix} \]3. **Calculating the Determinant of \(A\)**:
Use the determinant formula for a 3x3 matrix to compute \(\det(A)\). A non-zero determinant confirms there's a unique solution.4. **Creating Matrices \(A_x\), \(A_y\), \(A_z\)**:
Replace one column at a time of matrix \(A\) with the constants on the right side of the equations to form these matrices. Calculate their determinants as well.5. **Applying Cramer's Rule**:
Use the determinants to find the solutions: \[ x = \frac{\det(A_x)}{\det(A)}, \quad y = \frac{\det(A_y)}{\det(A)}, \quad z = \frac{\det(A_z)}{\det(A)} \]6. **Simplifying the Solutions**:
The final step is simplifying the fractions to obtain the clean set of values for the variables. This systematic approach ensures clarity and precision in solving such linear systems.