Problem 60 - Intermediate Algebra

Question

Use Cramer's rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. $$ \left\\{\begin{array}{l} 4 x+3 z=4 \\ 2 y-6 z=-1 \\ 8 x+4 y+3 z=9 \end{array}\right. $$

Step-by-Step Solution

Answer

Solution: \( x = \frac{1}{4}, \ y = \frac{35}{22}, \ z = \frac{1}{11} \).

Step 1: Set up the coefficient matrix

The first step in using Cramer's rule is to identify the coefficient matrix (denoted usually by \( A \)) that represents the system of equations. For the given system, the coefficient matrix \( A \) is determined by the coefficients of the variables \( x, y, \) and \( z \): \[A = \begin{bmatrix} 4 & 0 & 3 \0 & 2 & -6 \8 & 4 & 3 \end{bmatrix}\]

Step 2: Calculate the determinant of the coefficient matrix

Now that we have our coefficient matrix \( A \), we need to calculate its determinant, denoted \( \text{det}(A) \). Using the formula for a 3x3 determinant, compute:\[\text{det}(A) = 4(2 \cdot 3 - (-6) \cdot 4) - 0 + 3 (0 \cdot 3 - (-6) \cdot 8) = 4(6 + 24) + 3(0 + 48)\]After simplifying, we obtain:\[\text{det}(A) = 4 \cdot 30 + 3 \cdot 48 = 120 + 144 = 264\]

Step 3: Calculate the determinant for x (Dx)

Cramer's rule requires calculating the determinant \( D_x \) by replacing the first column of the coefficient matrix \( A \) with the constants from the right side of the equations. The matrix \( A_x \) then becomes:\[A_x = \begin{bmatrix} 4 & 0 & 3 \-1 & 2 & -6 \9 & 4 & 3 \end{bmatrix}\]Calculate \( \text{det}(A_x) \):\[\text{det}(A_x) = 4(2 \cdot 3 - (-6) \cdot 4) - (-1)(0 \cdot 3 - (-6) \cdot 9) + 3(0 \cdot 3 - 4 \cdot 9)\]Simplifying gives:\[\text{det}(A_x) = 4(6 + 24) + 9(6) - 3 \cdot 36 = 120 + 54 - 108 = 66\]

Step 4: Calculate the determinant for y (Dy)

For \( D_y \), substitute the second column with the constants:\[A_y = \begin{bmatrix} 4 & 4 & 3 \0 & -1 & -6 \8 & 9 & 3 \end{bmatrix}\]Calculate \( \text{det}(A_y) \):\[\text{det}(A_y) = 4(-1 \cdot 3 - (-6) \cdot 9) - 4(0 \cdot 3 - (-6) \cdot 8) + 3(0 \cdot 9 - (-1) \cdot 8)\]Simplifying gives:\[\text{det}(A_y) = 4(-3 + 54) + 4(0 + 48) + 3(0 + 8) = 204 + 192 + 24 = 420\]

Step 5: Calculate the determinant for z (Dz)

For \( D_z \), substitute the third column with the constants:\[A_z = \begin{bmatrix} 4 & 0 & 4 \0 & 2 & -1 \8 & 4 & 9 \end{bmatrix}\]Calculate \( \text{det}(A_z) \):\[\text{det}(A_z) = 4(2 \cdot 9 - 4 \cdot (-1)) - 0 + 4(0 \cdot 9 - 2 \cdot 8)\]Simplifying gives:\[\text{det}(A_z) = 4(18 + 4) - 4 \cdot 16 = 88 - 64 = 24\]

Step 6: Solve for the variables using Cramer's Rule

To find the values of \( x \), \( y \), and \( z \), divide each determinant for \( x \), \( y \), and \( z \) by \( \text{det}(A) \):\[x = \frac{D_x}{\text{det}(A)}, \quad y = \frac{D_y}{\text{det}(A)}, \quad z = \frac{D_z}{\text{det}(A)}\]Substitute:\[x = \frac{66}{264} = \frac{1}{4}, \quad y = \frac{420}{264} = \frac{35}{22}, \quad z = \frac{24}{264} = \frac{1}{11}\]

Step 7: Conclusion

The system of equations is consistent and independent, with unique solutions for each variable.

Key Concepts

System of Equations

A system of equations is simply a set of two or more equations with the same variables. The goal is to find values for the variables that satisfy all the equations simultaneously. In our exercise, we're dealing with three equations containing three variables: \(x\), \(y\), and \(z\). Solving such a system can tell us how these variables are interrelated.

• **Consistent**: A system is consistent if there's at least one set of variable values that satisfies all equations.
• **Inconsistent**: If no solution exists, the system is inconsistent.
• **Dependent**: A dependent system has infinitely many solutions, often due to the equations being multiples of one another.
• **Independent**: An independent system has exactly one solution, where each equation provides unique information about the relation of the variables.

By using mathematical methods such as substitution, graphing, or Cramer's rule, we can determine which case applies to our system.

Determinants

The determinant is a special number that you can calculate from a square matrix. It's a crucial tool when dealing with system of equations, especially in matrix algebra.
In our exercise, each solution requires calculating the determinant of a matrix formed from the coefficients of our system of equations.
For a 3x3 matrix like our coefficient matrix \( A \), the determinant can be calculated as follows:

• Multiply diagonals: Start by multiplying the elements across the diagonals of the matrix in two directions.
• Subtract: Subtract the sum of products of one set of diagonal lines from the other.

Having a non-zero determinant signifies that the system has a unique solution. When the determinant of the coefficient matrix is zero, it typically indicates that the system is either inconsistent or dependent, meaning alternative solutions or no solutions exist. This is why calculating the determinants of the transformed matrices is vital in the process.

Matrix Algebra

Matrix algebra is a powerful mathematical framework used to handle and manipulate matrices effectively. It lays down a foundation for working with systems of equations and many other mathematical domains.
For the given system, we utilize several matrices:

• **Coefficient Matrix**: Represents the coefficients of the variables in the system. This matrix is denoted by \( A \).
• **Determinants in Matrices**: Each variable's solution is connected to a specific determinant, which is manipulated by replacing columns of the coefficient matrix with solution constants.
• **Applying Cramer's Rule**: Once the determinants are calculated, Cramer's Rule gives us a method to find the variables with simple divisions.

By treating these equations with matrix operations, we can scale up solutions efficiently, even when the equations become complex. Matrix operations allow us to systematically find and verify solutions while maintaining clear steps, as demonstrated when working through the original exercise.