Problem 42 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 33-42: Use Cramer's rule, whenever applicable, to solve the system. $$ \left\\{\begin{aligned} 4 x-y+3 z &=6 \\ -8 x+3 y-5 z &=-6 \\ 5 x-4 y &=-9 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solution is \( x = 1 \), \( y = -12.75 \), \( z = -3 \).

Step 1: Write the System of Equations in Matrix Form

The system of equations can be represented in matrix form as \( AX = B \) where \( A \) is the coefficient matrix, \( X \) is the column matrix of variables \( x, y, z \), and \( B \) is the column matrix of constants:\[A = \begin{bmatrix} 4 & -1 & 3 \ -8 & 3 & -5 \ 5 & -4 & 0 \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad B = \begin{bmatrix} 6 \ -6 \ -9 \end{bmatrix}\]

Step 2: Find the Determinant of Matrix A

The determinant of matrix \( A \), \( \det(A) \), is calculated as follows:\[\det(A) = \begin{vmatrix} 4 & -1 & 3 \ -8 & 3 & -5 \ 5 & -4 & 0 \end{vmatrix}\]By expanding along the first row, we get:\[= 4 \begin{vmatrix} 3 & -5 \ -4 & 0 \end{vmatrix} - (-1) \begin{vmatrix} -8 & -5 \ 5 & 0 \end{vmatrix} + 3 \begin{vmatrix} -8 & 3 \ 5 & -4 \end{vmatrix} \]Calculating each of the smaller determinants:\[= 4(3 \cdot 0 - (-5) \cdot (-4)) + 1((-8 \cdot 0) - (-5) \cdot 5) + 3((-8 \cdot -4) - 5 \cdot 3) \]\[= 4(0 - 20) + 1(0 - (-25)) + 3(32 - 15) \]\[= -80 + 25 + 51 \]\[= -4\]Thus, \( \det(A) = -4 eq 0 \), confirming that Cramer's rule applies.

Step 3: Replace Columns to Find Determinants for Cramer's Rule

To find \( x, y, z \) using Cramer's rule, compute the determinants of matrices formed by replacing the corresponding columns of \( A \) with \( B \).**For \( x \):**\( A_x = \begin{bmatrix} 6 & -1 & 3 \ -6 & 3 & -5 \ -9 & -4 & 0 \end{bmatrix} \)\[\det(A_x) = \begin{vmatrix} 6 & -1 & 3 \ -6 & 3 & -5 \ -9 & -4 & 0 \end{vmatrix} = -4 \; \text{(as previously calculated directly for } x \text{)}\]**For \( y \):**\( A_y = \begin{bmatrix} 4 & 6 & 3 \ -8 & -6 & -5 \ 5 & -9 & 0 \end{bmatrix} \)\[\det(A_y) = \begin{vmatrix} 4 & 6 & 3 \ -8 & -6 & -5 \ 5 & -9 & 0 \end{vmatrix} = 51\]**For \( z \):**\( A_z = \begin{bmatrix} 4 & -1 & 6 \ -8 & 3 & -6 \ 5 & -4 & -9 \end{bmatrix} \)\[\det(A_z) = \begin{vmatrix} 4 & -1 & 6 \ -8 & 3 & -6 \ 5 & -4 & -9 \end{vmatrix} = 12\]

Step 4: Calculate Variables Using Cramer's Rule

Cramer's Rule provides:\[x = \frac{\det(A_x)}{\det(A)} = \frac{-4}{-4} = 1\]\[y = \frac{\det(A_y)}{\det(A)} = \frac{51}{-4} = -12.75\]\[z = \frac{\det(A_z)}{\det(A)} = \frac{12}{-4} = -3\]

Step 5: Solution

The solution for the system of equations is \( x = 1 \), \( y = -12.75 \), and \( z = -3 \).

Key Concepts

System of Equations

A system of equations consists of two or more equations with the same set of unknowns. In mathematics, systems of equations are used to find multiple variables at once. For example, when you have a set of equations, like the one in the original exercise, each equation is a relationship between several variables, namely \(x\), \(y\), and \(z\).
These equations can be connected to real-world situations, such as where these variables could represent quantities like distances, speeds, or costs in different scenarios. Solving such a system of equations means finding a specific set of values for the variables that make all the given equations true simultaneously.
There are several methods to solve a system of equations, including substitution, elimination, and matrix methods like Cramer's Rule.

• Substitution involves solving one equation for one variable and then using that solution to find the other variables.
• Elimination adds or subtracts equations to cancel out one of the variables, allowing an easier solution.
• Matrix methods, such as Cramer's Rule, use linear algebra concepts to find the solution more efficiently for larger systems.

Understanding these methods will help you to choose the most effective one based on the number of equations and their complexity.

Determinant of a Matrix

The determinant of a matrix is a special number calculated from its elements. In a system of equations represented in matrix form, the determinant can tell us important properties about the matrix, such as whether it is invertible or has a unique solution.
The determinant, noted as \( \det(A) \) for a matrix \(A\), is a scalar value that often appears in square matrices, i.e., matrices with an equal number of rows and columns. To calculate the determinant of a 3x3 matrix, you can use the cofactor expansion method.
This expansion involves selecting a row or column, multiplying each element by the determinant of the smaller 2x2 matrix that remains after removing the row and column containing the element, and combining these in a specific way. The result is crucial when applying Cramer's Rule, as a non-zero determinant indicates that the system of equations has a unique solution.

• For example, in the given exercise, the determinant \( \det(A) = -4 \), which is not zero, confirms that the equations have a viable, single solution.
• If \( \det(A) \) was zero, it would suggest the system might have infinitely many solutions or no solution at all.

Computing determinants is a fundamental skill in algebra for solving systems efficiently.

Matrices in Algebra

Matrices are a rectangular arrangement of numbers into rows and columns and are powerful tools in algebra. They provide a compact way of managing and computing complex systems, like systems of equations. In algebra, matrices can represent anything from simple mappings to complex transformations.
In the context of solving systems of linear equations, matrices help us organize the coefficients and the constants into structured forms. This is seen in the exercise where we use matrices \( A \), \( X \), and \( B \) as representations of coefficients, variables, and constants.
By transforming a system of equations into a matrix form \(AX = B\), you can apply various matrix operations, such as finding determinants or inverses, to solve for the variable matrix \(X\).
Matrices allow operations that would be cumbersome to perform with raw numbers, such as:

• Addition or subtraction of systems, which parallels elimination in algebra.
• Multiplication, which can represent applying the same operation across many equations simultaneously.
• Inverses, which help find solutions by effectively 'dividing' by the coefficient matrix.

Understanding how to use matrices in algebra accelerates the process of solving equations and broadens the scope to solve complex and high-dimensional problems efficiently.