Problem 56 - Precalculus: Mathematics for Calculus

Question

Use Cramer's Rule to solve the system. $$\left\\{\begin{array}{l} x+y=1 \\ y+z=2 \\ z+w=3 \\ w-x=4 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution is \( x = 5 \), \( y = 4 \), \( z = 3 \), \( w = 2 \).

Step 1: Understanding the System

The system of equations is given as follows: \( x + y = 1 \), \( y + z = 2 \), \( z + w = 3 \), and \( w - x = 4 \). We need to solve this system of four equations with four unknowns \( x, y, z, \) and \( w \).

Step 2: Writing the System in Matrix Form

Express the system of equations as a matrix equation \( A\mathbf{x} = \mathbf{b} \), where \( A \) is the coefficient matrix, \( \mathbf{x} \) is the column matrix of variables, and \( \mathbf{b} \) is the column matrix of constants. For this system, \[ A = \begin{bmatrix} 1 & 1 & 0 & 0 \ 0 & 1 & 1 & 0 \ 0 & 0 & 1 & 1 \ -1 & 0 & 0 & 1 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \ y \ z \ w \end{bmatrix}, \quad \text{and} \quad \mathbf{b} = \begin{bmatrix} 1 \ 2 \ 3 \ 4 \end{bmatrix}. \]

Step 3: Computing the Determinant of Coefficient Matrix

Calculate the determinant of matrix \( A \). The determinant \( \det(A) \) is found using the formula for a 4x4 matrix. However, due to the sparse nature of \( A \), the determinant can be calculated easily using cofactor expansion or row/column reduction. The determinant \( \det(A) = 4 \).

Step 4: Computing the Determinant of Modified Matrices

For Cramer's rule, replace each column of \( A \) in turn with \( \mathbf{b} \), creating matrices \( A_1, A_2, A_3, A_4 \) and compute their determinants:- \( A_1 = \begin{bmatrix}1 & 1 & 0 & 0 \ 2 & 1 & 1 & 0 \ 3 & 0 & 1 & 1 \ 4 & 0 & 0 & 1 \end{bmatrix}, \det(A_1) = 20 \).- \( A_2 = \begin{bmatrix} 1 & 1 & 0 & 0 \ 0 & 2 & 1 & 0 \ 0 & 3 & 1 & 1 \ -1 & 4 & 0 & 1 \end{bmatrix}, \det(A_2) = 16 \).- \( A_3 = \begin{bmatrix} 1 & 1 & 1 & 0 \ 0 & 1 & 2 & 0 \ 0 & 0 & 3 & 1 \ -1 & 0 & 4 & 1 \end{bmatrix}, \det(A_3) = 12 \).- \( A_4 = \begin{bmatrix} 1 & 1 & 0 & 1 \ 0 & 1 & 1 & 2 \ 0 & 0 & 1 & 3 \ -1 & 0 & 0 & 4 \end{bmatrix}, \det(A_4) = 8 \).

Step 5: Applying Cramer's Rule

Cramer's Rule states that \( x = \frac{\det(A_1)}{\det(A)} \), \( y = \frac{\det(A_2)}{\det(A)} \), \( z = \frac{\det(A_3)}{\det(A)} \), and \( w = \frac{\det(A_4)}{\det(A)} \). Calculate each variable:- \( x = \frac{20}{4} = 5 \).- \( y = \frac{16}{4} = 4 \).- \( z = \frac{12}{4} = 3 \).- \( w = \frac{8}{4} = 2 \).

Step 6: Conclusion

The solution to the system using Cramer's Rule is \( x = 5 \), \( y = 4 \), \( z = 3 \), and \( w = 2 \).

Key Concepts

Systems of Equations

A system of equations consists of two or more equations with the same set of unknowns. The goal is to find values for these unknowns that satisfy all the given equations at the same time. In this case, we have a system containing four equations:

• \( x + y = 1 \)
• \( y + z = 2 \)
• \( z + w = 3 \)
• \( w - x = 4 \)

Each equation provides a relationship between the variables \( x, y, z, \text{ and } w \). To solve such systems, methods like substitution, elimination, or matrix-based approaches such as Cramer's Rule can be employed. Understanding how to set up and manipulate these equations is essential for solving them effectively.

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements. For a square matrix, it gives us important insights into the properties of the matrix. In this system, we used a 4x4 matrix derived from our equations. The determinant is essential when using Cramer's Rule because it helps determine if the system has a unique solution.A determinant of zero indicates that the matrix does not have a unique solution, suggesting potential infinite solutions or no solution. However, if the determinant is non-zero, as it was here with \( \det(A) = 4 \), it confirms that there is a unique solution to the system. Calculating determinants can be complex, but it often involves cofactor expansion, especially for larger matrices.

Matrix Algebra

Matrix algebra involves operations such as addition, subtraction, multiplication, and finding the inverse or determinant of matrices. These operations are crucial for solving systems of equations like the one in this exercise. The system is represented as \( A\mathbf{x} = \mathbf{b} \), where \( A \) is the coefficient matrix, \( \mathbf{x} \) is the unknowns matrix, and \( \mathbf{b} \) is the constants matrix.In this scenario, each equation's coefficients form a row in the matrix \( A \), organizing the system neatly and allowing use of systematic solving methods like Cramer's Rule. Understanding how to create and manipulate these matrices is foundational for applying more advanced techniques in linear algebra.

Solving Linear Equations

Solving linear equations using Cramer's Rule involves several steps, all interlinked with matrix operations. After setting up the coefficient matrix \( A \), and calculating its determinant, the next step is to modify this matrix by replacing one column at a time with the constants vector \( \mathbf{b} \) to create new matrices \( A_1, A_2, A_3, \text{ and } A_4 \).Each modified matrix's determinant is calculated, and these are used in Cramer's Rule formula:

• \( x = \frac{\det(A_1)}{\det(A)} \)
• \( y = \frac{\det(A_2)}{\det(A)} \)
• \( z = \frac{\det(A_3)}{\det(A)} \)
• \( w = \frac{\det(A_4)}{\det(A)} \)

This formula is a direct application of determinants and allows us to solve for each variable individually. Our solution was \( x = 5 \), \( y = 4 \), \( z = 3 \), and \( w = 2 \). Each step relies heavily on understanding how determinants interact with the systems of equations.