Problem 45 - Precalculus

Question

Apply Cramer's rule to solve each system of equations, if possible. $$\begin{array}{r} 3 x+8 y+2 z=28 \\ -2 x+5 y+3 z=34 \\ 4 x+9 y+2 z=29 \end{array}$$

Step-by-Step Solution

Answer

The solution is approximately \( x \approx 3.24 \), \( y \approx 1.06 \), \( z \approx 4.53 \).

Step 1: Identify the Coefficient Matrix

First, identify the coefficient matrix from the given system of equations: \[A = \begin{bmatrix} 3 & 8 & 2 \ -2 & 5 & 3 \ 4 & 9 & 2 \end{bmatrix}\] The coefficient matrix \( A \) contains the coefficients of the variables \( x, y, z \) from each equation.

Step 2: Calculate the Determinant of A

Calculate the determinant of the matrix \( A \) using the formula for the determinant of a 3x3 matrix: \[det(A) = 3(5 \cdot 2 - 3 \cdot 9) - 8(-2 \cdot 2 - 3 \cdot 4) + 2(-2 \cdot 9 - 5 \cdot 4)\]Simplify to find:\( det(A) = 3(-17) - 8(8) + 2(-38) = -51 - 64 - 76 = -191 \).

Step 3: Cramer's Rule Setup

Cramer's rule states that for the system \( Ax = b \), each variable \( x_i \) can be found using \( x_i = \frac{det(A_i)}{det(A)} \), where \( A_i \) is the matrix formed by replacing the \( i^{th} \) column of \( A \) with the column matrix \( b = \begin{bmatrix} 28 \ 34 \ 29 \end{bmatrix} \).

Step 4: Calculate Determinant for \( x \)

Form the matrix \( A_1 \) by replacing the first column of \( A \) with \( b \):\[ A_1 = \begin{bmatrix} 28 & 8 & 2 \ 34 & 5 & 3 \ 29 & 9 & 2 \end{bmatrix} \]Calculate \( det(A_1) \):\[ det(A_1) = 28(5 \cdot 2 - 3 \cdot 9) - 8(34 \cdot 2 - 3 \cdot 29) + 2(34 \cdot 9 - 5 \cdot 29) \]Calculate this expression: \( det(A_1) = 28(-17) - 8(38) + 2(81) = -476 - 304 + 162 = -618 \).

Step 5: Calculate Determinant for \( y \)

Form the matrix \( A_2 \) by replacing the second column of \( A \) with \( b \):\[ A_2 = \begin{bmatrix} 3 & 28 & 2 \ -2 & 34 & 3 \ 4 & 29 & 2 \end{bmatrix} \]Calculate \( det(A_2) \):\[ det(A_2) = 3(34 \cdot 2 - 3 \cdot 29) - 28(-2 \cdot 2 - 3 \cdot 4) + 2(-2 \cdot 29 - 34 \cdot 4) \]Calculate this expression:\( det(A_2) = 3(11) + 28(4) + 2(-174) = 33 + 112 - 348 = -203 \).

Step 6: Calculate Determinant for \( z \)

Form the matrix \( A_3 \) by replacing the third column of \( A \) with \( b \):\[ A_3 = \begin{bmatrix} 3 & 8 & 28 \ -2 & 5 & 34 \ 4 & 9 & 29 \end{bmatrix} \]Calculate \( det(A_3) \):\[ det(A_3) = 3(5 \cdot 29 - 34 \cdot 9) - 8(-2 \cdot 29 - 34 \cdot 4) + 28(-2 \cdot 9 - 5 \cdot 4) \]Calculate this expression:\( det(A_3) = 3(-211) + 8(174) + 28(-58) = -633 + 1392 - 1624 = -865 \).

Step 7: Applying Cramer's Rule to Solve for Variables

Using the determinants calculated, apply Cramer's rule:- \( x = \frac{det(A_1)}{det(A)} = \frac{-618}{-191} = \ 3.23508 \)- \( y = \frac{det(A_2)}{det(A)} = \frac{-203}{-191} = \ 1.06283 \)- \( z = \frac{det(A_3)}{det(A)} = \frac{-865}{-191} = \ 4.52775 \)

Key Concepts

Determinant Calculation

Calculating the determinant of a matrix is a pivotal step in using Cramer's Rule to solve systems of equations. The determinant gives us a unique scalar value that provides insights into the properties of the matrix. For a 3x3 matrix, the determinant is computed using a specific formula:

• Multiply the diagonal elements of the matrix, and use combinations of minors and cofactors.
• Expand along any row or column, but traditionally, we expand along the first row for simplicity.

The determinant provides valuable information:

• If the determinant is non-zero, the matrix has an inverse, which means the system of equations has a unique solution.
• If the determinant is zero, the system may be either inconsistent (no solutions) or dependent (infinitely many solutions).

In the exercise example, calculating the determinant of matrix \( A \) revealed the value is \(-191\), indicating a unique solution exists.

Systems of Equations

A system of equations consists of two or more equations with the same set of variables, such as \( x, y, \) and \( z \). Solving these systems involves finding the values of these variables that satisfy all given equations simultaneously. There are several methods for solving systems of equations, such as:

• Graphical method - Plotting the equations and finding intersection points.
• Substitution - Solving one equation for a variable and substituting it into the other equation.
• Elimination - Adding or subtracting equations to eliminate one variable, then solving the reduced system.
• Matrix methods - Using techniques like Cramer's Rule or Gaussian elimination.

In the exercise, the system is solved using Cramer's Rule, a method facilitated by determinants and particularly effective when the number of equations matches the number of variables.

Linear Algebra

Linear algebra is a branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vectors. It provides tools for modeling and solving real-world problems. Important concepts include:

• Vectors - Objects that have both a direction and a magnitude.
• Matrices - Rectangular arrays of numbers with operations similar to numerical operations.
• Transformations - Functions that map vectors to other vectors using matrices.
• Determinants and eigenvalues - Scalars that reveal matrix properties and system stability.

In the context of this exercise, linear algebra allows us to represent the system of equations as a matrix equation \( Ax = b \), simplifying the computational process and enabling the use of Cramer's Rule for efficient problem-solving.

Matrix Algebra

Matrix algebra involves the manipulation of matrices to perform various operations, including addition, subtraction, multiplication, and finding inverses. These operations are essential in solving systems of equations, transforming geometric spaces, and modeling complex phenomena. Key operations include:

• Matrix multiplication - Combining matrices to form new matrices, akin to combining function compositions.
• Matrix inversion - Finding a matrix that reverses matrix multiplication's effects, similar to division in arithmetic.
• Using matrices to shift and scale spaces in transformations.

Cramer's Rule, as used in the exercise, is deeply rooted in matrix algebra. It leverages the matrix's structure to isolate each variable's value. The replacement of matrix columns to form \( A_i \) matrices shows the powerful application of matrix operations to evaluate solutions deterministically by computing determinants rather than grappling with substitutions or graphical intercepts.