Problem 66 - A Graphical Approach to College Algebra

Question

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{array}{l}5 x+2 y+z=15 \\\2 x-y+z=9 \\\4 x+3 y+2 z=13\end{array}$$

Step-by-Step Solution

Answer

The solution is \( x = 3 \), \( y = -1 \), \( z = 2 \).

Step 1: Write the System of Equations in Matrix Form

First, identify the coefficients of the variables and write the system in matrix form. The given system of equations is:\[\begin{align*}5x + 2y + z &= 15 \2x - y + z &= 9 \4x + 3y + 2z &= 13\end{align*}\]This corresponds to matrix form \( AX = B \):\[ A = \begin{bmatrix} 5 & 2 & 1 \ 2 & -1 & 1 \ 4 & 3 & 2 \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad B = \begin{bmatrix} 15 \ 9 \ 13 \end{bmatrix} \]

Step 2: Calculate the Determinant of Matrix A (Denoted as D)

To use Cramer's rule, first calculate the determinant of matrix \( A \):\[D = \begin{vmatrix} 5 & 2 & 1 \ 2 & -1 & 1 \ 4 & 3 & 2 \end{vmatrix}\]Apply the formula for determinant of a 3x3 matrix:\[ D = 5(-1 \cdot 2 - 1 \cdot 3) - 2(2 \cdot 2 - 1 \cdot 4) + 1(2 \cdot 3 - (-1) \cdot 4) \]\[ D = 5(-2 - 3) - 2(4 - 4) + 1(6 + 4) \]\[ D = 5(-5) - 2(0) + 1(10) \]\[ D = -25 + 0 + 10 = -15 \]

Step 3: Determine the Cramer's Rule Numerators (D_x, D_y, D_z)

With \( D eq 0 \), calculate the determinants for \( D_x \), \( D_y \), and \( D_z \):For \( D_x \), replace the first column of \( A \) with \( B \):\[D_x = \begin{vmatrix} 15 & 2 & 1 \ 9 & -1 & 1 \ 13 & 3 & 2 \end{vmatrix}\]\[ D_x = 15(-1 \cdot 2 - 1 \cdot 3) - 2(9 \cdot 2 - 1 \cdot 13) + 1(9 \cdot 3 - (-1) \cdot 13) \]\[ D_x = 15(-5) - 2(18 - 13) + 1(27 + 13) \]\[ D_x = -75 - 10 + 40 = -45 \]For \( D_y \), replace the second column with \( B \):\[D_y = \begin{vmatrix} 5 & 15 & 1 \ 2 & 9 & 1 \ 4 & 13 & 2 \end{vmatrix}\]\[ D_y = 5(9 \cdot 2 - 1 \cdot 13) - 15(2 \cdot 2 - 1 \cdot 4) + 1(2 \cdot 13 - 9 \cdot 4) \]\[ D_y = 5(18 - 13) - 15(4 - 4) + 1(26 - 36) \]\[ D_y = 25 + 0 - 10 = 15 \]Lastly, for \( D_z \), replace the third column with \( B \):\[D_z = \begin{vmatrix} 5 & 2 & 15 \ 2 & -1 & 9 \ 4 & 3 & 13 \end{vmatrix}\]\[ D_z = 5(-1 \cdot 13 - 9 \cdot 3) - 2(2 \cdot 13 - 9 \cdot 4) + 15(2 \cdot 3 - (-1) \cdot 4) \]\[ D_z = 5(-13 - 27) - 2(26 - 36) + 15(6 + 4) \]\[ D_z = 5(-40) + 2(10) + 15(10) \]\[ D_z = -200 + 20 + 150 = -30 \]

Step 4: Apply Cramer's Rule to Find the Solutions

Now, use Cramer's rule to find \( x \), \( y \), and \( z \):\[ x = \frac{D_x}{D} = \frac{-45}{-15} = 3 \]\[ y = \frac{D_y}{D} = \frac{15}{-15} = -1 \]\[ z = \frac{D_z}{D} = \frac{-30}{-15} = 2 \]Thus, the solution is \( x = 3 \), \( y = -1 \), and \( z = 2 \).

Key Concepts

Determinant

A determinant is a special number that can be calculated from a square matrix. It provides important information about the matrix, such as whether the matrix is invertible or not. For a 3x3 matrix like the one given in the exercise, the determinant is calculated using a specific formula that involves the elements of the matrix. In this case, the determinant helps to verify that the set of equations in question can indeed be solved using Cramer's Rule.

To calculate the determinant of matrix \( A \), we use the formula:

• Find the product of the diagonal elements.
• Subtract the products of the elements that create "cross" patterns.
• Combine these components according to their positions in the formula.

The determinant of matrix \( A \) is calculated as follows: \[ D = a(ei − fh) − b(di − fg) + c(dh − eg) \] For the given matrix: \[ D = 5(-1 \cdot 2 - 1 \cdot 3) - 2(2 \cdot 2 - 1 \cdot 4) + 1(2 \cdot 3 - (-1) \cdot 4) = -15 \] The determinant \( D \) is found to be \( -15 \). Since it is not zero, it allows us to proceed with Cramer's Rule to solve the equations.

System of Equations

A system of equations consists of multiple equations that have common variables. The goal is to find values for these variables that satisfy all of the equations simultaneously. In the exercise, we're dealing with three equations and three variables:\( x, y, \text{and } z \). This is a typical linear system of equations.

The given system:

• \(5x + 2y + z = 15\)
• \(2x - y + z = 9\)
• \(4x + 3y + 2z = 13\)

The Cramer's Rule is applicable because the number of equations is equal to the number of variables. This method specifically requires a non-zero determinant, which confirms that a unique solution exists for this system.

When a system cannot be solved directly due to a zero determinant, other methods such as substitution, elimination, or using an inverse matrix may be helpful. Hence, determining the determinable nature of the system through testing the determinant is an essential first step.

Matrix Form

Representing a system of equations in matrix form is a valuable method that simplifies complex systems and facilitates the use of techniques like Cramer's Rule. By arranging coefficients in a matrix, the structure of the equations becomes clear and manageable. Matrix form succinctly expresses the connections and relationships between variables and constants in a system of equations.

In this exercise, the system is rewritten using matrices as:

• \( A = \begin{bmatrix} 5 & 2 & 1 \ 2 & -1 & 1 \ 4 & 3 & 2 \end{bmatrix} \)
• \( X = \begin{bmatrix} x \ y \ z \end{bmatrix} \)
• \( B = \begin{bmatrix} 15 \ 9 \ 13 \end{bmatrix} \)

This allows for systematic analysis by aligning equations with the row and column framework of matrices. It simplifies solving by enabling us to employ linear algebra techniques, providing a structured pathway to finding the values of \( x, y, \text{and } z \). This approach underscores the profound utility of matrices in mathematics, especially in managing multi-variable linear equations.