Problem 69 - A Graphical Approach to Precalculus with Limits

Question

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{array}{l}5 x-y=-4 \\\3 x+2 z=4 \\\4 y+3 z=22\end{array}$$

Step-by-Step Solution

Answer

Solution: \(x = 0\), \(y = 4\), \(z = 2\).

Step 1: Write the System in Matrix Form

Write the system of equations as a matrix equation of the form \(A\mathbf{x} = \mathbf{b}\), where \(A\) is the coefficient matrix, \(\mathbf{x}\) is the vector of variables, and \(\mathbf{b}\) is the vector of constants. For the given system:\[A = \begin{bmatrix} 5 & -1 & 0 \ 3 & 0 & 2 \ 0 & 4 & 3 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} -4 \ 4 \ 22 \end{bmatrix}\]

Step 2: Calculate the Determinant of the Coefficient Matrix

Find the determinant \(D\) of the coefficient matrix \(A\):\[D = \begin{vmatrix} 5 & -1 & 0 \ 3 & 0 & 2 \ 0 & 4 & 3 \end{vmatrix}\]Using cofactor expansion along the first row:\[D = 5 \begin{vmatrix} 0 & 2 \ 4 & 3 \end{vmatrix} - (-1) \begin{vmatrix} 3 & 2 \ 0 & 3 \end{vmatrix} + 0 \begin{vmatrix} 3 & 0 \ 0 & 4 \end{vmatrix}\]\[= 5 (0 \cdot 3 - 2 \cdot 4) + 1 (3 \cdot 3 - 2 \cdot 0)\]\[= 5(0 - 8) + 1(9 - 0)\]\[= 5(-8) + 9\]\[= -40 + 9 = -31\]

Step 3: Confirm Determinant is Non-zero

Since \(D = -31\), the determinant is non-zero, which means we can use Cramer's rule to find the solution.

Step 4: Apply Cramer's Rule to Find \(x\)

Replace the first column of the matrix \(A\) with the vector \(\mathbf{b}\) to find \(D_x\):\[A_x = \begin{bmatrix} -4 & -1 & 0 \ 4 & 0 & 2 \ 22 & 4 & 3 \end{bmatrix}\]Calculate \(D_x\):\[D_x = \begin{vmatrix} -4 & -1 & 0 \ 4 & 0 & 2 \ 22 & 4 & 3 \end{vmatrix}\]Using cofactor expansion along the first row:\[D_x = -4 \begin{vmatrix} 0 & 2 \ 4 & 3 \end{vmatrix} - (-1) \begin{vmatrix} 4 & 2 \ 22 & 3 \end{vmatrix} + 0\]\[= -4(0 \cdot 3 - 2 \cdot 4) + (4 \cdot 3 - 2 \cdot 22)\]\[= -4(0 - 8) + (12 - 44)\]\[= 32 + (-32) = 0\]Thus, \(x = \frac{D_x}{D} = \frac{0}{-31} = 0\)

Step 5: Apply Cramer's Rule to Find \(y\)

Replace the second column of the matrix \(A\) with the vector \(\mathbf{b}\) to find \(D_y\):\[A_y = \begin{bmatrix} 5 & -4 & 0 \ 3 & 4 & 2 \ 0 & 22 & 3 \end{bmatrix}\]Calculate \(D_y\):\[D_y = \begin{vmatrix} 5 & -4 & 0 \ 3 & 4 & 2 \ 0 & 22 & 3 \end{vmatrix}\]Using cofactor expansion along the first row:\[D_y = 5 \begin{vmatrix} 4 & 2 \ 22 & 3 \end{vmatrix} - (-4) \begin{vmatrix} 3 & 2 \ 0 & 3 \end{vmatrix} + 0\]\[= 5(4 \cdot 3 - 2 \cdot 22) + 4(9)\]\[= 5(12 - 44) + 36\]\[= 5(-32) + 36\]\[= -160 + 36 = -124\]Thus, \(y = \frac{D_y}{D} = \frac{-124}{-31} = 4\)

Step 6: Apply Cramer's Rule to Find \(z\)

Replace the third column of the matrix \(A\) with the vector \(\mathbf{b}\) to find \(D_z\):\[A_z = \begin{bmatrix} 5 & -1 & -4 \ 3 & 0 & 4 \ 0 & 4 & 22 \end{bmatrix}\]Calculate \(D_z\):\[D_z = \begin{vmatrix} 5 & -1 & -4 \ 3 & 0 & 4 \ 0 & 4 & 22 \end{vmatrix}\]Using cofactor expansion along the first row:\[D_z = 5 \begin{vmatrix} 0 & 4 \ 4 & 22 \end{vmatrix} - (-1) \begin{vmatrix} 3 & 4 \ 0 & 22 \end{vmatrix} + (-4) \begin{vmatrix} 3 & 0 \ 0 & 4 \end{vmatrix}\]\[= 5(0 \cdot 22 - 4 \cdot 4) + 1(3 \cdot 22 - 4 \cdot 0) - 4(12)\]\[= 5(0 - 16) + 66 - 48\]\[= -80 + 66 - 48\]\[= -62\]Thus, \(z = \frac{D_z}{D} = \frac{-62}{-31} = 2\)

Step 7: Solution Interpretation

The solution of the system using Cramer's rule is \(x = 0, y = 4, z = 2\). All determinants were calculated correctly and the solution is consistent with the original equation.

Key Concepts

Determinant Calculation

The determinant of a matrix is a special number that provides important information about the matrix's properties. For example, if the determinant of the matrix is zero, it indicates that the system of equations has no unique solutions.

When dealing with a 3x3 matrix, the determinant can be calculated using cofactor expansion. This involves choosing a row or column and performing calculations on the minors of each element. The minor is the determinant of the 2x2 submatrix that remains when you exclude the row and column of the specific element you are considering.

For the given coefficient matrix:

• Choose an element from the first row, calculate its minor, and alternate signs starting from positive.
• The calculation involves taking each element of the row, multiplying it by its cofactor (the minor), and summing up these products.

The calculated determinant being non-zero means we can proceed to use Cramer's rule to find the solution to the system of equations.

Matrix Equation

Writing the system of equations in matrix form is a useful step that allows us to handle the equations in a more structured way. Matrix equations are written in the form \(A\mathbf{x} = \mathbf{b}\), where \(A\) is the coefficient matrix containing all the coefficients of the variables, \(\mathbf{x}\) is a column matrix (also called a vector) of the variables, and \(\mathbf{b}\) is a column matrix of constants.

In the given system, the matrix representation is:

• \(A = \begin{bmatrix} 5 & -1 & 0 \ 3 & 0 & 2 \ 0 & 4 & 3 \end{bmatrix}\)
• \(\mathbf{x} = \begin{bmatrix} x \ y \ z \end{bmatrix}\)
• \(\mathbf{b} = \begin{bmatrix} -4 \ 4 \ 22 \end{bmatrix}\)

This setup helps in using determinant calculation methods like Cramer's rule, as it allows us to work algebraically with the matrices to find the variable values.

System of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. In this exercise, we have three equations with three unknowns - \(x\), \(y\), and \(z\). Each equation represents a plane in a three-dimensional space.

To solve a system of equations using Cramer's Rule:

• Ensure the number of equations matches the number of unknowns for a square system, which can potentially have a unique solution.
• Convert the system into a matrix equation so you can apply matrix algebra techniques.

For the given system, the solutions are determined by analyzing how the planes intersect. If there is a unique intersection point, the system has a unique solution, as shown by a non-zero determinant.

Cofactor Expansion

Cofactor expansion is a method used to calculate the determinant of a matrix. This process involves breaking down the determinant calculation into smaller, more manageable parts called minors.

For a 3x3 matrix, a cofactor expansion is typically performed along a row or a column:

• Select the row (often the first row is chosen to simplify calculations).
• For each element in the chosen row, calculate the minor - the determinant of the 2x2 matrix that remains after removing the selected element's row and column.
• Multiply each element by its cofactor, which combines its minor and a sign factor that alternates between positive and negative as you move across the row.

This method is particularly useful because it reduces the complexity of the determinant calculation by focusing on a series of smaller, 2x2 determinations.