Problem 52 - Intermediate Algebra

Question

Use Cramer's rule to solve each system of equations. $$ \left\\{\begin{array}{l} x+y+z=4 \\ x-y+z=2 \\ x-y-z=0 \end{array}\right. $$

Step-by-Step Solution

Answer

The solutions are \(x = 4\), \(y = 1\), and \(z = 0\).

Step 1: Write the System in Matrix Form

Convert the system of linear equations into the matrix form.The given system:1. \(x + y + z = 4\)2. \(x - y + z = 2\)3. \(x - y - z = 0\)Can be written in matrix form as:\[\begin{bmatrix}1 & 1 & 1 \1 & -1 & 1 \1 & -1 & -1 \end{bmatrix}\begin{bmatrix}x \ y \ z \end{bmatrix} = \begin{bmatrix}4 \ 2 \ 0 \end{bmatrix}\]Here, \(A\) is the coefficient matrix, \([x \, y \, z]^T\) is the variable matrix, and the right-hand side constants are in the matrix on the right.

Step 2: Calculate the Determinant of the Coefficient Matrix

Find the determinant of the coefficient matrix \(A\):\[A = \begin{bmatrix}1 & 1 & 1 \1 & -1 & 1 \1 & -1 & -1 \end{bmatrix}\]The determinant \(\det(A)\) is calculated as follows:\[\det(A) = 1((-1)(-1) - 1(-1)) - 1(1(-1) - 1) + 1(1(-1) - (-1))\]\[\det(A) = 1(1 + 1) - 1(-1 - 1) + 1(-1 + 1)\]\[\det(A) = 2 + 2 + 0 = 4\]

Step 3: Solve for x Using Cramer's Rule

Substitute the first column of \(A\) with the constants to form a new matrix \(A_x\):\[A_x = \begin{bmatrix}4 & 1 & 1 \2 & -1 & 1 \0 & -1 & -1 \end{bmatrix}\]Calculate \(\det(A_x)\):\[\det(A_x) = 4((-1)(-1) - 1(-1)) - 1(2(-1) - 1 \times 0) + 1(2(-1) - (-1) \times 0)\]\[\det(A_x) = 4(1 + 1) + 2 + 2 = 16\]Then, use Cramer's Rule:\[x = \frac{\det(A_x)}{\det(A)} = \frac{16}{4} = 4\]

Step 4: Solve for y Using Cramer's Rule

Substitute the second column of \(A\) with the constants to form \(A_y\):\[A_y = \begin{bmatrix}1 & 4 & 1 \1 & 2 & 1 \1 & 0 & -1 \end{bmatrix}\]Calculate \(\det(A_y)\):\[\det(A_y) = 1(2(-1) - 1 \times 0) - 4(1(-1) - 1) + 1(1 \times 0 - 2 \times 1)\]\[\det(A_y) = 1(-2) + 4(1 + 1) - 2 = -2 + 8 - 2 = 4\]Using Cramer's Rule:\[y = \frac{\det(A_y)}{\det(A)} = \frac{4}{4} = 1\]

Step 5: Solve for z Using Cramer's Rule

Substitute the third column of \(A\) with the constants to form \(A_z\):\[A_z = \begin{bmatrix}1 & 1 & 4 \1 & -1 & 2 \1 & -1 & 0 \end{bmatrix}\]Calculate \(\det(A_z)\):\[\det(A_z) = 1((-1)(0) - 2(-1)) - 1(1 \times 0 - 2 \times 1) + 4(1(-1) - (-1))\]\[\det(A_z) = 1(0 + 2) + 1(0 - 2) + 4(-1 + 1)\]\[\det(A_z) = 2 - 2 + 0 = 0\]Using Cramer's Rule:\[z = \frac{\det(A_z)}{\det(A)} = \frac{0}{4} = 0\]

Key Concepts

Systems of Equations

A system of equations is a collection of two or more equations with a common set of variables. In this exercise, we have three linear equations involving the variables \(x\), \(y\), and \(z\). Solving such systems allows us to find the values for each variable that satisfy all the given equations simultaneously. Systems of equations can be solved using various methods, such as substitution, elimination, and matrix methods like Cramer's Rule. Understanding these equations is crucial as they form the basis for complex problems in science and engineering. By representing these systems in different forms, we can analyze and solve them more efficiently.

Determinants

Determinants are a special number that can be calculated from a square matrix. They hold important properties and provide useful information about the matrix, such as whether it is invertible. In Cramer's Rule, we use determinants to find solutions to systems of linear equations. For example, when given the matrix \(A\) from the coefficient part of our system, calculating its determinant helps us determine whether a unique solution exists. If the determinant is zero, the system could be dependent or inconsistent. Calculating determinants involves specific arithmetic operations on the elements of the matrix, which can be simplified using properties such as minors and cofactors.

Linear Algebra

Linear algebra is a powerful field of mathematics that deals with vectors, vector spaces, and linear mappings. It provides a framework for solving systems of linear equations, which is essential for various applications in physics, computer science, and engineering. Concepts like matrices and determinants are fundamental to linear algebra and play a key role in solutions like Cramer's Rule. By studying linear algebra, we gain a deeper understanding of how to work with multidimensional data and perform operations that lead to meaningful solutions in real-world problems.

Matrix Form

In solving systems of equations, representing the equations in matrix form simplifies the process. A matrix organizes the coefficients of the variables compactly, allowing us to use matrix operations for solutions. In our exercise, the system of equations is transformed into a matrix form where each equation corresponds to a row in a matrix, and each variable is represented in a separate column. By using matrices, we can efficiently apply methods such as Gaussian elimination or, as shown here, Cramer's Rule. This representation is crucial for computational purposes and helps visualize the structure of the equations.