Problem 23 - Algebra for College Students

Question

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{c}2 x-y+3 z=-17 \\ 3 y+z=5 \\ x-2 y-z=-3\end{array}\right)\)

Step-by-Step Solution

Answer

The solution is approximately \(x \approx 8.21\), \(y \approx 2.43\), \(z \approx -9.86\).

Step 1: Write Augmented Matrix of the System

First, we represent the given system of equations in matrix form. The coefficient matrix \(A\) and the constant matrix \(B\) are initialized with respective coefficients and constants:\[ A = \begin{bmatrix} 2 & -1 & 3 \ 0 & 3 & 1 \ 1 & -2 & -1 \end{bmatrix}, \ B = \begin{bmatrix} -17 \ 5 \ -3 \end{bmatrix} \]

Step 2: Check the Determinant of Matrix A

Calculate the determinant of matrix \(A\), \(\Delta\). If \(\Delta = 0\), the system might be dependent or inconsistent.\[ \Delta = \begin{vmatrix} 2 & -1 & 3 \ 0 & 3 & 1 \ 1 & -2 & -1 \end{vmatrix} = 2(3(-1) - 1(-2)) - (-1)(0(-1) - 1(1)) + 3(0(-2) - 3(1)) = -4 - 1 - 9 = -14 \]Since \(\Delta = -14 eq 0\), the system is independent, and a unique solution exists.

Step 3: Calculate Determinant for \(x\) (\(\Delta_x\))

Replace the first column of \(A\) with \(B\) to form \(A_x\) and calculate its determinant:\[ A_x = \begin{bmatrix} -17 & -1 & 3 \ 5 & 3 & 1 \ -3 & -2 & -1 \end{bmatrix} \]\[ \Delta_x = \begin{vmatrix} -17 & -1 & 3 \ 5 & 3 & 1 \ -3 & -2 & -1 \end{vmatrix} = -17(3(-1) - 1(-2)) + 1(-(-3)(1) - 1(5)) + 3(5(-2) - 3(3)) = -60 + 8 - 63 = -115 \]

Step 4: Calculate Determinant for \(y\) (\(\Delta_y\))

Replace the second column of \(A\) with \(B\) to form \(A_y\) and calculate its determinant:\[ A_y = \begin{bmatrix} 2 & -17 & 3 \ 0 & 5 & 1 \ 1 & -3 & -1 \end{bmatrix} \]\[ \Delta_y = \begin{vmatrix} 2 & -17 & 3 \ 0 & 5 & 1 \ 1 & -3 & -1 \end{vmatrix} = 2(5(-1) - 1(-3)) - (-17)(0*(-1) - 1(1)) + 3(0(-3) - 1(5)) = -20 + 6 - 5 - 15 = -34 \]

Step 5: Calculate Determinant for \(z\) (\(\Delta_z\))

Replace the third column of \(A\) with \(B\) to form \(A_z\) and calculate its determinant:\[ A_z = \begin{bmatrix} 2 & -1 & -17 \ 0 & 3 & 5 \ 1 & -2 & -3 \end{bmatrix} \]\[ \Delta_z = \begin{vmatrix} 2 & -1 & -17 \ 0 & 3 & 5 \ 1 & -2 & -3 \end{vmatrix} = 2(3(-3) - 5(-2)) + 1(0(-3) - 5(1)) - 17(0(-2) - 3(3)) = -10 - 5 + 153 = 138 \]

Step 6: Solve for Variables Using Cramer's Rule

Now we apply Cramer's Rule to find the values of \(x\), \(y\), and \(z\):\[ x = \frac{\Delta_x}{\Delta} = \frac{-115}{-14} = \frac{115}{14} \approx 8.2143 \]\[ y = \frac{\Delta_y}{\Delta} = \frac{-34}{-14} = \frac{34}{14} \approx 2.4286 \]\[ z = \frac{\Delta_z}{\Delta} = \frac{138}{-14} = -\frac{138}{14} = -\frac{69}{7} \approx -9.8571 \]

Key Concepts

Determinant

The determinant is a special number that can be calculated from a square matrix. It is a crucial concept in linear algebra, particularly when solving systems of linear equations using methods such as Cramer's Rule. To solve a system of equations using determinants, one must calculate the determinant of the coefficient matrix. If this determinant is zero, the system may not have a unique solution — it might be dependent or inconsistent. The determinant gives insight into the nature of the system:

• A non-zero determinant indicates the matrix is invertible, and the system of equations has a unique solution.
• A zero determinant suggests either dependency or inconsistencies within the system, meaning solutions could be infinite or nonexistent.

Calculating a determinant involves expanding along a row or column and following specific arithmetic rules. For a 3x3 matrix, you usually expand the determinant by minors, involving several multiplication and addition operations. The sign changes for each element according to its position.

Linear Equations

Linear equations form the foundation of algebra and involve expressions that set variables to first-degree polynomial equations. These equations take the form of lines when graphed and are crucial for many calculations in real-world applications. For our exercise, we have a system of linear equations that can be expressed in the following standard form: \[ ax + by + cz = d \] where \(x\), \(y\), and \(z\) are the variables, and \(a\), \(b\), \(c\), and \(d\) are constants.

Solving these linear equations often involves finding the point(s) where these lines intersect. There are various methods to determine this, such as substitution, elimination, or graphical methods. However, when using matrix algebra and determinants, Cramer's Rule becomes especially useful; it provides an efficient way to solve these systems mathematically.

Matrix Algebra

Matrix algebra is a branch of mathematics focusing on square and rectangular arrays of numbers called matrices. It includes operations like addition, multiplication, and matrix inversion. In the context of solving linear equations, it's an efficient means to represent systems compactly and to perform several operations simultaneously.

A system of linear equations can be represented using matrices, thus facilitating ease of calculation using rules of matrix algebra. The equations' coefficients are placed into a so-called coefficient matrix. With Cramer's Rule, we also create other matrices by replacing one column at a time with the constant terms from the equations' right-hand sides, computing their determinants, and then using these to solve for the unknowns.

• The coefficient matrix contains only the coefficients of the variables.
• Matrix inversion, which is linked closely with determinants, opens many paths for finding solutions.

Independent System

In the world of systems of equations, an independent system is one where the equations intersect at exactly one single point. This means there is exactly one solution for the set of equations, indicating that the equations are truly independent of one another. For a system to be independent and have a unique solution, the determinant of the coefficient matrix must be non-zero.

Systems can be classified into:

• Independent systems, which have unique solutions.
• Dependent systems, which have infinite solutions because the equations represent the same line or plane.
• Inconsistent systems, which have no solution because the equations do not intersect at any point.

In the problem we've been solving, calculating the determinant of the coefficient matrix revealed a non-zero result, confirming that the system is independent. Thus, the use of Cramer's Rule gives us a singular solution for each variable.