Problem 30 - Algebra and Trigonometry

Question

Use Cramer's Rule to solve each system. $$\left\\{\begin{aligned}x-y+2 z &=3 \\\2 x+3 y+z &=9 \\\\-x-y+3 z &=11\end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \( x = -1/3, y = -2, z = 3 \).

Step 1: Write the Coefficient Matrix and Calculate Its Determinant

First, form the matrix \( A \) using the coefficients of \( x \), \( y \), and \( z \) in the three equations. So, \( A = \[ \[1, -1, 2], [2, 3, 1], [-1, -1, 3] \] \). Then, calculate \( det(A) = 1*(3*3 - 1*(-1)) - (-1)*(2*3 - 1*(-1)) + 2*(2*(-1) - 1*-3) = 18.

Step 2: Form the Modified Matrices and Calculate Their Determinants

Now in order to calculate the value of each variable, replace the column of coefficients for that variable in matrix A with the constants from the right side of the equations to form \( A_x \), \( A_y \), \( A_z \). Then, calculate the determinants of these matrices using the same method as in step 1. This gives \( det(A_x) = -6 \), \( det(A_y) = -36 \), and \( det(A_z) = 54 \).

Step 3: Apply Cramer's Rule

Cramer's Rule states that the solution to the system is \( x = det(A_x) / det(A) \), \( y = det(A_y) / det(A) \), and \( z = det(A_z) / det(A) \). Substituting the determinants calculated in Step 2, we get \( x = -6 / 18 = -1/3 \), \( y = -36 / 18 = -2 \), and \( z = 54 / 18 = 3 \).

Key Concepts

Systems of Linear Equations

When dealing with systems of linear equations, you're working with multiple equations that you solve simultaneously. Each equation typically involves the same set of variables. In our exercise, we have three equations with three unknowns—specifically, variables \( x \), \( y \), and \( z \). The goal is to find a set of values for these variables that satisfy all equations at once.

The given system is:
\[\begin{aligned} x-y+2z &=3, \ 2x+3y+z &=9, \ -x-y+3z &=11. \end{aligned}\]

The key to solving these equations is recognizing that they represent planes in a three-dimensional space. Where these planes intersect is the solution to the system. The method we use, Cramer's Rule, is particularly useful for solving systems where the number of equations matches the number of unknowns. It leverages determinants to find the values that satisfy all equations simultaneously.

Determinants

A determinant is a unique numerical value that can be calculated from the elements of a square matrix. It provides insights into the nature of the matrix and solutions of the system of equations. When solving systems of linear equations with Cramer's Rule, determinants play a crucial role. They help in finding the values of variables directly, given that the determinant of the coefficient matrix is non-zero.

In our example, the determinant of the coefficient matrix \( A \) was calculated to be 18. This non-zero determinant indicates that the system of equations has a unique solution. If it were zero, the system might be either dependent or inconsistent, representing no solution or infinitely many solutions respectively.
The determinants of modified matrices (\( A_x \), \( A_y \), and \( A_z \)) are used to specifically calculate each variable using Cramer’s Rule.

Matrix Algebra

Matrix algebra is a mathematical framework used to work with collections of numbers (matrices) that model and solve systems involving linear equations. In this context, you deal with matrices by performing operations like addition, multiplication, and finding determinants. These operations can simplify the process of solving complex systems of equations.

For instance, the coefficient matrix \( A \) for the given equation system is:
\[\begin{bmatrix} 1 & -1 & 2 \ 2 & 3 & 1 \ -1 & -1 & 3 \end{bmatrix}\]

Matrix algebra allows you to manipulate this matrix to extract solutions. By modifying certain columns with constants (`3`, `9`, `11`) and computing their determinants, you can systematically solve the entire set of equations. This illustrates how matrices are not just collections of numbers, but powerful tools in solving linear equations.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations and their representations through matrices and vector spaces. It encompasses the theories and techniques used to solve systems of linear equations, analyze transformations, and perform operations involving matrices.

Understanding linear algebra concepts such as linear independence, span, and basis help in grasping deeper insights into solutions of linear systems. In our system, we apply linear algebra principles to understand how equations are represented in a matrix form and solved using determinants.

Cramer's Rule, used in this exercise, is a technique derived from linear algebra that employs determinants to solve for each variable. This rule reveals the interconnectedness of linear algebra concepts like matrices and determinants with practical problem-solving methods in real-world applications.