Problem 17 - 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. \(\left(\begin{array}{rl}x-y+2 z & =-8 \\ 2 x+3 y-4 z & =18 \\ -x+2 y-z & =7\end{array}\right)\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{116}{15}, \, y = \frac{20}{3}, \, z = \frac{3}{5}\).

Step 1: Write the Matrix Representation

First, we need to express the given system of equations in a matrix form. The system is: \(x - y + 2z = -8\), \(2x + 3y - 4z = 18\), and \(-x + 2y - z = 7\). The matrix equation is represented as \(AX = B\) where:\[A = \begin{bmatrix} 1 & -1 & 2 \ 2 & 3 & -4 \ -1 & 2 & -1 \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad B = \begin{bmatrix} -8 \ 18 \ 7 \end{bmatrix}\]

Step 2: Find the Determinant of A

To use Cramer's rule, calculate the determinant of matrix \(A\), denoted as \( \det(A) \). The determinant of a \(3 \times 3\) matrix is found using:\[\det(A) = \left| \begin{matrix} 1 & -1 & 2 \ 2 & 3 & -4 \ -1 & 2 & -1 \end{matrix} \right|= 1(3 - 8) + 1(2 + 4) + 2(4 + 3)= -5 + 6 + 14 = 15\]Thus, \(\det(A) = 15\).

Step 3: Determine if the System is Solvable

Since \(\det(A) = 15\), which is not zero, the system is independent, and a unique solution exists.

Step 4: Use Cramer's Rule to Find x

To find \(x\), replace the first column of \(A\) with \(B\) and calculate the determinant:\[A_x = \begin{bmatrix} -8 & -1 & 2 \ 18 & 3 & -4 \ 7 & 2 & -1 \end{bmatrix} \]\[\det(A_x) = -8(3 - 8) + 1(18 + 28) + 2(36 - 21)= 40 + 46 + 30 = 116\]Then, compute \(x\):\[x = \frac{\det(A_x)}{\det(A)} = \frac{116}{15}\]

Step 5: Use Cramer's Rule to Find y

To find \(y\), replace the second column of \(A\) with \(B\) and calculate the determinant:\[A_y = \begin{bmatrix} 1 & -8 & 2 \ 2 & 18 & -4 \ -1 & 7 & -1 \end{bmatrix} \]\[\det(A_y) = 1(-18 + 28) + 8(2 + 4) + 2(14 + 7)= 10 + 48 + 42 = 100\]Then, compute \(y\):\[y = \frac{\det(A_y)}{\det(A)} = \frac{100}{15}\]

Step 6: Use Cramer's Rule to Find z

To find \(z\), replace the third column of \(A\) with \(B\) and calculate the determinant:\[A_z = \begin{bmatrix} 1 & -1 & -8 \ 2 & 3 & 18 \ -1 & 2 & 7 \end{bmatrix}\]\[\det(A_z) = 1(21 - 36) + 1(-14 - 18) + 8(4 + 3)= -15 - 32 + 56 = 9\]Then, compute \(z\):\[z = \frac{\det(A_z)}{\det(A)} = \frac{9}{15}\]

Step 7: Simplify the Fraction Results

Convert the obtained fractions to their simplest forms:- \(x = \frac{116}{15}\) is already in simplest form.- \(y = \frac{100}{15} = \frac{20}{3}\).- \(z = \frac{9}{15} = \frac{3}{5}\).Thus, the solution of the system is:\(x = \frac{116}{15}, \, y = \frac{20}{3}, \, z = \frac{3}{5}\).

Key Concepts

Systems of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. In the provided exercise, we have a system of three equations involving three variables: \(x\), \(y\), and \(z\). The goal is to find values for these variables that satisfy all equations simultaneously. Systems of equations can come in different forms:

• Independent Systems: These have a unique solution, meaning one set of values for \(x\), \(y\), and \(z\) that satisfy all the equations.
• Dependent Systems: These have infinitely many solutions. In this case, all the equations represent the same plane in space.
• Inconsistent Systems: These have no solution, often because the equations represent parallel planes that never intersect.

In our case, because the determinant of the matrix (as we see in the solution) is not zero, the system is independent. This implies that a unique solution exists. Thus, learning to identify the type of system is crucial when solving equations.

Matrix Algebra

Matrix algebra involves a set of mathematical operations and principles that help in the manipulation and solution of matrices. In solving systems of equations with matrix representation, we express the system using matrices, seen in the form \(AX = B\):

• \(A\) is the coefficient matrix comprising the coefficients of the variables.
• \(X\) is the column matrix of the variables, \([x \ y \ z]^T\).
• \(B\) is the constant matrix containing the constants from the equations' right-hand side.

In the given exercise, matrix \(A\) is constructed using the coefficients of the variables from each equation, and this structure simplifies complex computations like finding determinants or using Cramer's Rule. Understanding the notation and structure allows operations like matrix multiplication and row transformation, which are essential for solving such problems. Matrix algebra provides a solid framework for handling and solving systems with multiple equations.

Determinants of Matrices

The determinant of a matrix is a special number that provides important information about the matrix. For a \(3 \times 3\) matrix, it is calculated using a specific formula that involves the elements of the matrix. Here’s what the determinant helps with:

• Determining Solvability: If the determinant of the coefficient matrix \(A\) is zero, the system of equations either has no solution or has infinitely many solutions.
• Calculations in Cramer's Rule: To use Cramer's Rule effectively, calculating the determinant is a crucial step, as it helps facilitate the determination of each variable.

In our solution, the determinant of matrix \(A\) was calculated as \(15\), which is non-zero. This confirms the system is solvable with a unique solution. The determinant plays a vital role in many areas of linear algebra, offering insights not only into solution existence and uniqueness but also in transformations and changes to space described by the matrix.