Problem 10 - Advanced Engineering Mathematics

Question

Solve the given system of equations by Cramer's rule. $$ \begin{aligned} 4 x+3 y+2 z &=8 \\ -x+& 2 z=12 \\ 3 x+2 y+z &=3 \end{aligned} $$

Step-by-Step Solution

Answer

The solution is \( x = -2 \), \( y = 2 \), \( z = -229 \).

Step 1: Identify the System of Equations

The system of equations given is:1. \( 4x + 3y + 2z = 8 \)2. \( -x + 0y + 2z = 12 \)3. \( 3x + 2y + z = 3 \).These equations need to be solved using Cramer's Rule.

Step 2: Form the Coefficient Matrix

Prepare the coefficient matrix \( A \) from the system of equations:\[A = \begin{pmatrix}4 & 3 & 2 \-1 & 0 & 2 \3 & 2 & 1\end{pmatrix}.\]

Step 3: Calculate the Determinant of the Coefficient Matrix

Calculate \( \det(A) \) using the formula for a 3x3 matrix determinant:\[\det(A) = 4(0 \cdot 1 - 2 \cdot 2) - 3(-1 \cdot 1 - 2 \cdot 3) + 2(-1 \cdot 2 - 3 \cdot 0).\]Simplifying gives:\[\det(A) = 4(0 - 4) + 3(1 + 6) + 2(-2 - 0) = -16 + 21 - 4 = 1.\]

Step 4: Create Matrix for \( x \) and Calculate its Determinant

Create matrix \( A_x \) by replacing the first column of \( A \) with the constant terms:\[A_x = \begin{pmatrix}8 & 3 & 2 \12 & 0 & 2 \3 & 2 & 1\end{pmatrix}.\]Calculate \( \det(A_x) \):\[\det(A_x) = 8(0 - 2 \cdot 2) - 3(12 \cdot 1 - 2 \cdot 3) + 2(12 \cdot 2 - 3 \cdot 0).\]Simplifying gives:\[\det(A_x) = 8(0 - 4) - 3(12 - 6) + 2(24 - 0) = -32 - 18 + 48 = -2.\]

Step 5: Create Matrix for \( y \) and Calculate its Determinant

Create matrix \( A_y \) by replacing the second column of \( A \) with the constant terms:\[A_y = \begin{pmatrix}4 & 8 & 2 \-1 & 12 & 2 \3 & 3 & 1\end{pmatrix}.\]Calculate \( \det(A_y) \):\[\det(A_y) = 4(12 \cdot 1 - 2 \cdot 3) - 8(-1 \cdot 1 - 2 \cdot 3) + 2(-1 \cdot 3 - 12 \cdot 3).\]Simplifying gives:\[\det(A_y) = 4(12 - 6) - 8(1 + 6) + 2(-3 - 36) = 24 + 56 - 78 = 2.\]

Step 6: Create Matrix for \( z \) and Calculate its Determinant

Create matrix \( A_z \) by replacing the third column of \( A \) with the constant terms:\[A_z = \begin{pmatrix}4 & 3 & 8 \-1 & 0 & 12 \3 & 2 & 3\end{pmatrix}.\]Calculate \( \det(A_z) \):\[\det(A_z) = 4(0 \cdot 3 - 2 \cdot 12) - 3(-1 \cdot 3 - 12 \cdot 3) + 8(-1 \cdot 2 - 3 \cdot 0).\]Simplifying gives:\[\det(A_z) = 4(0 - 24) - 3(3 + 36) + 8(-2) = -96 - 117 - 16 = -229.\]

Step 7: Solve for \( x \), \( y \), \( z \) using Cramer's Rule

Apply Cramer's Rule to solve for \( x \), \( y \), and \( z \):\[x = \frac{\det(A_x)}{\det(A)} = \frac{-2}{1} = -2, \y = \frac{\det(A_y)}{\det(A)} = \frac{2}{1} = 2, \z = \frac{\det(A_z)}{\det(A)} = \frac{-229}{1} = -229.\]

Step 8: Final Solution of the System

The solution to the system of equations using Cramer's Rule is \( x = -2 \), \( y = 2 \), and \( z = -229 \).

Key Concepts

System of Equations

A system of equations consists of two or more equations set to be satisfied simultaneously. Each equation represents a relationship involving variables, commonly shown as unknown elements such as \( x \), \( y \), and \( z \). In our example, the system of equations is expressed as three equations:

• \( 4x + 3y + 2z = 8 \)
• \( -x + 2z = 12 \)
• \( 3x + 2y + z = 3 \)

The goal is to find values for \( x \), \( y \), and \( z \) that satisfy all equations simultaneously. Each equation can be visualized as a plane in a 3-dimensional space, and the solution represents the intersection point of these planes. Cramer's Rule offers an elegant approach to solve such systems when the number of equations equals the number of unknowns, provided the system has a unique solution.

Determinants

Determinants are a property of square matrices, which are fundamental in solving systems of linear equations using Cramer's Rule. They provide a scalar value that represents the matrix, allowing the assessment of system uniqueness and solvability.For a 3x3 matrix as mentioned in the exercise, the determinant \( \det(A) \) is calculated through the formula:\[\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]where \( a, b, c, d, e, f, g, h, \) and \( i \) are elements of the matrix.In our exercise, calculating the determinant of the coefficient matrix indicates whether the system has a single unique solution. If \( \det(A) eq 0 \), the system is invertible, and Cramer's Rule can be used effectively to solve for unknown variables. Thus, determinants are instrumental in determining system characteristics and applying matrix operations.

Matrix Algebra

Matrix algebra plays a crucial role in solving systems of equations, particularly when employing Cramer's Rule. Matrices are structured arrays of numbers arranged in rows and columns, used to organize coefficients and constants systematically from the equations.In Cramer's Rule, the approach involves forming different matrices:

• The **coefficient matrix** \( A \) represents the coefficients of the variables.
• **Modified matrices** (like \( A_x, A_y, A_z \)) are used, where one column is replaced by the constants from the equations.

Matrix operations such as calculating determinants allow us to solve for each variable by replacing columns, maintaining an organized system for algebraic manipulation.Matrix algebra not only enables a simplified visualization of complex systems but also ensures efficient computations through defined rules and operations, making it a powerful tool in computational mathematics.