Problem 46 - Precalculus

Question

Apply Cramer's rule to solve each system of equations, if possible. $$\begin{aligned} 7 x+2 y-z &=-1 \\ 6 x+5 y+z &=16 \\ -5 x-4 y+3 z &=-5 \end{aligned}$$

Step-by-Step Solution

Answer

The solution is \(x = -1, y = 4, z = \frac{69}{86}\).

Step 1: Write the System of Equations in Matrix Form

Convert the given system of equations into matrix form. The system is given by \( AX = B \), where \( A \) is the coefficient matrix, \( X \) is the variable matrix, and \( B \) is the constant matrix. \[ A = \begin{bmatrix} 7 & 2 & -1 \ 6 & 5 & 1 \ -5 & -4 & 3 \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad B = \begin{bmatrix} -1 \ 16 \ -5 \end{bmatrix} \]

Step 2: Calculate Determinant of Coefficient Matrix \(A\)

Using the coefficient matrix \( A \), find the determinant \( \text{det}(A) \). The determinant for a 3x3 matrix is calculated as:\[ \text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \]Substitute the corresponding values:\[ = 7(5 \times 3 - 1 \times (-4)) - 2(6 \times 3 - 1 \times (-5)) - 1(6 \times (-4) - 5 \times (-5)) \]\[ = 7(15 + 4) - 2(18 + 5) - 1(-24 + 25) \]\[ = 7 \times 19 - 2 \times 23 - 1 \times 1 \]\[ = 133 - 46 - 1 \]\[ = 86 \]Since \( \text{det}(A) eq 0 \), Cramer's Rule can be applied.

Step 3: Calculate Determinant for \(x\) using \( A_x \)

Replace the first column of matrix \( A \) with matrix \( B \) to form \( A_x \).\[ A_x = \begin{bmatrix} -1 & 2 & -1 \ 16 & 5 & 1 \ -5 & -4 & 3 \end{bmatrix} \]Calculate \( \text{det}(A_x) \) using the same formula:\[ = -1(5 \times 3 - 1 \times (-4)) - 2(16 \times 3 - 1 \times (-5)) - 1(16 \times (-4) - 5 \times (-5)) \]\[ = -1(15 + 4) - 2(48 + 5) - 1(-64 + 25) \]\[ = -1 \times 19 - 2 \times 53 + 1 \times 39 \]\[ = -19 - 106 + 39 \]\[ = -86 \]

Step 4: Calculate Determinant for \(y\) using \( A_y \)

Replace the second column of matrix \( A \) with matrix \( B \) to form \( A_y \).\[ A_y = \begin{bmatrix} 7 & -1 & -1 \ 6 & 16 & 1 \ -5 & -5 & 3 \end{bmatrix} \]Calculate \( \text{det}(A_y) \):\[ = 7(16 \times 3 - 1 \times (-5)) + 1(6 \times 3 - 1 \times (-5)) - 1(6 \times (-5) - 16 \times (-5)) \]\[ = 7(48 + 5) + 1(18 + 5) - 1(-30 + 80) \]\[ = 7 \times 53 + 1 \times 23 - 1 \times 50 \]\[ = 371 + 23 - 50 \]\[ = 344 \]

Step 5: Calculate Determinant for \(z\) using \( A_z \)

Replace the third column of matrix \( A \) with matrix \( B \) to form \( A_z \).\[ A_z = \begin{bmatrix} 7 & 2 & -1 \ 6 & 5 & 16 \ -5 & -4 & -5 \end{bmatrix} \]Calculate \( \text{det}(A_z) \):\[ = 7(5 \times (-5) - 16 \times (-4)) - 2(6 \times (-5) - 16 \times (-5)) + 1(6 \times (-4) - 5 \times 16) \]\[ = 7(-25 + 64) - 2(-30 + 80) + 1(-24 - 80) \]\[ = 7 \times 39 - 2 \times 50 - 1 \times 104 \]\[ = 273 - 100 - 104 \]\[ = 69 \]

Step 6: Solve for \(x, y, z\) using Cramer's Rule

Now use Cramer's Rule to find the variables:\[ x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{-86}{86} = -1 \]\[ y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{344}{86} = 4 \]\[ z = \frac{\text{det}(A_z)}{\text{det}(A)} = \frac{69}{86} = \frac{69}{86} \, (simplified) \]

Key Concepts

Determinant Calculation

The concept of determinant plays a crucial role when solving systems of linear equations using Cramer's Rule, especially for 3x3 matrices. A determinant is a special number that can be calculated from the elements of a square matrix. It provides insight into the matrix's properties, such as whether it is invertible or not.

For a 3x3 matrix, the determinant is calculated using the rule of Sarrus or by specific formula: \[ \text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg). \] Here, each term involves the permutation of elements from rows and columns of the matrix. You replace one element from each row and each column in a specific way. This process might seem tedious at first glance but becomes intuitive with practice.

Calculating the determinant tells us whether a set of linear equations has a unique solution. If the determinant of the coefficient matrix is zero, the system may either have no solutions or infinitely many solutions. If it is not zero, we can proceed with methods like Cramer's Rule to find a unique solution.

3x3 System of Equations

A 3x3 system of equations consists of three linear equations that we solve simultaneously to find the values of the three variables that satisfy all equations. These systems are compactly represented in matrix form to simplify both representation and computation. This involves three key components:

• The coefficient matrix \(A\), where each element corresponds to a coefficient in one of the linear equations.
• The variable matrix \(X\), which contains the variables we are solving for, for example, \(x, y,\) and \(z\).
• The constant matrix \(B\), made up of the constants from the equations' right-hand side.

Representing the equations in matrix form allows us to apply algebraic operations that are easier to manage, especially for larger systems. This abstraction makes techniques like Cramer's Rule or Gaussian elimination possible, aiding in finding solutions more efficiently than handling each equation separately.

Matrix Method

The matrix method is a powerful tool for solving linear equations. It leverages matrix operations, which are systematic and efficient, to find solutions to linear systems. Using matrices, we can easily manipulate equations and apply rules like Cramer's Rule to solve for unknowns.

In the given problem, the system of equations is represented as \(AX=B\). Here, matrix \(A\) comprises the coefficients, \(X\) is the variable matrix \([x, y, z]^T\), and \(B\) is the constants matrix. By expressing the system in this form, we can systematically calculate each variable using determinants:

• Replace a column of \(A\) with \(B\) to form \(A_x\), \(A_y\), or \(A_z\), depending on the variable to solve.
• Compute the determinant for these new matrices.
• Use Cramer's Rule: \( x = \frac{\text{det}(A_x)}{\text{det}(A)} \), \( y = \frac{\text{det}(A_y)}{\text{det}(A)} \), \( z = \frac{\text{det}(A_z)}{\text{det}(A)} \).

This procedure, while formulaic, is straightforward once you become familiar with matrix operations and determinant calculations. It's an organized approach, saving time and effort compared to solving equations manually one by one.