Problem 22 - Algebra 2

Question

Use Cramer’s Rule to solve each system of equations. \(4 x+2 y-3 z=-32\) \(-x-3 y+z=54\) \(2 y+8 z=78\)

Step-by-Step Solution

Answer

The solutions are \( x = 2 \), \( y = -20 \), \( z = 9 \) using Cramer's Rule.

Step 1: Write the System in Matrix Form

First, we need to represent the given system of equations in matrix form as \( AX = B \), where \( A \) is the coefficient matrix, \( X \) is the column matrix of variables, and \( B \) is the column matrix of constants. For our equations, we have:\[A = \begin{bmatrix} 4 & 2 & -3 \ -1 & -3 & 1 \ 0 & 2 & 8 \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad B = \begin{bmatrix} -32 \ 54 \ 78 \end{bmatrix}\]

Step 2: Calculate the Determinant of Matrix A (Det A)

The determinant of matrix \( A \) is found using the formula for the determinant of a 3x3 matrix:\[\text{det}(A) = \begin{vmatrix} 4 & 2 & -3 \ -1 & -3 & 1 \ 0 & 2 & 8 \end{vmatrix} = 4(-3 \cdot 8 - 2 \cdot 1) - 2(-1 \cdot 8 - 0 \cdot 1) - 3(-1 \cdot 2 - 0 \cdot -3)\]After calculation, we get:\[\text{det}(A) = 4(-24 - 2) + 2(-8) + 3(-2) = 4(-26) - 16 - 6 = -104 - 16 - 6 = -126\]

Step 3: Calculate Determinants for Each Variable

For each variable, replace the respective column in \( A \) with \( B \) and find the determinant:For \( x \):\[A_x = \begin{bmatrix} -32 & 2 & -3 \ 54 & -3 & 1 \ 78 & 2 & 8 \end{bmatrix}, \quad \text{det}(A_x) = -32((-3 \cdot 8 - 2 \cdot 1)) - 2(54 \cdot 8 - 1 \cdot 78) - 3(54 \cdot 2 - (-3) \cdot 78)\]Calculate \( \text{det}(A_x) \).For \( y \):\[A_y = \begin{bmatrix} 4 & -32 & -3 \ -1 & 54 & 1 \ 0 & 78 & 8 \end{bmatrix}, \quad \text{det}(A_y) = 4(54 \cdot 8 - 1 \cdot 78) - (-32)(-1 \cdot 8) + (-3)((-1) \cdot 78)\]Calculate \( \text{det}(A_y) \).For \( z \):\[A_z = \begin{bmatrix} 4 & 2 & -32 \ -1 & -3 & 54 \ 0 & 2 & 78 \end{bmatrix}, \quad \text{det}(A_z) = 4((-3) \cdot 78 - 54 \cdot 2) + 2((-1) \cdot 78) - 32((-1) \cdot 2)\]Calculate \( \text{det}(A_z) \).

Step 4: Solve for Each Variable Using Cramer's Rule

Using Cramer's Rule, each variable is calculated using the formula \( x = \frac{\text{det}(A_x)}{\text{det}(A)} \), \( y = \frac{\text{det}(A_y)}{\text{det}(A)} \), and \( z = \frac{\text{det}(A_z)}{\text{det}(A)} \).Substitute the determinants calculated into these formulas for \( x \), \( y \), and \( z \), respectively.

Key Concepts

Determinant of a Matrix

A determinant of a matrix is a special scalar value that can be calculated from the elements of a square matrix. It's pivotal in solving systems of linear equations using Cramer's Rule and has considerable importance in linear algebra. The determinant helps in understanding whether a matrix has an inverse, and in this context, it's vital for evaluating whether a unique solution exists for the system of equations.

For a 3x3 matrix, the determinant can be calculated using a specific formula:

• Start by selecting any row or column (commonly the first row or column).
• Multiply each element by the determinant of the 2x2 matrix that remains after removing the row and column of that element.
• Add or subtract these products according to a fixed pattern of signs: \(+, -, +\) for the first, second, and third positions respectively if working row-wise.

In the given problem, the determinant of the coefficient matrix \(A\) is calculated as \(-126\), which is crucial because a non-zero determinant indicates the system has a unique solution.

System of Equations

A system of equations is a collection of two or more equations with a similar set of unknowns. Solving these systems means finding the set of values for the variables that satisfies all the equations simultaneously.

The given problem involves three equations with three unknowns: \(x\), \(y\), and \(z\). Solving this system required using Cramer's Rule, which applies well when the number of equations equals the number of unknowns and the determinant of the coefficient matrix is non-zero.

This ensures each variable can be solved distinctively, by isolating it in terms of a determinant ratio. The approach demonstrates the alignment of each equation towards a common solution point, especially when translated into a matrix form—highlighting intersecting planes in geometric interpretation.

Matrix Algebra

Matrix algebra involves various mathematical operations and concepts applied to matrices. These operations include addition, subtraction, and multiplication of matrices, but also the concept of matrix inversion and determinants.

In the context of solving systems of equations, matrix algebra provides a systematic method:

• Transform the system of equations into a matrix form \(AX = B\), where \(A\) represents the coefficients, \(X\) the variables, and \(B\) the constants.
• Utilize the properties of matrices, specifically determinants, to apply methods like Cramer's Rule.
• The operations reflect transformations of space, showing how matrix coefficients interact geometrically.

Our example illustrates the elegance of matrix algebra in simplifying potentially complex systems into manageable linear transformations. This is why understanding matrices deeply enriches one's comprehension of linear equations and their solutions across varied applications.