Problem 38 - College Algebra with Corequisite Support

Question

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{r} 4 x-3 y+4 z=10 \\ 5 x-2 z=-2 \\ 3 x+2 y-5 z=-9 \end{array} $$

Step-by-Step Solution

Answer

The solution is \(x = 60\), \(y = -122\), \(z = -269\).

Step 1: Identify Coefficients and Constants

First, we need to identify the coefficients of the variables and the constant terms from the system of equations. The given system can be written in matrix form as:\[A = \begin{bmatrix}4 & -3 & 4 \5 & 0 & -2 \3 & 2 & -5\end{bmatrix}, \quad x = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad B = \begin{bmatrix} 10 \ -2 \ -9 \end{bmatrix}\]

Step 2: Find the Determinant of Matrix A

Calculate the determinant of matrix \(A\) using the formula for a 3x3 matrix determinant:\[\det(A) = 4(0 \cdot -5 - (-2) \cdot 2) - (-3)(5 \cdot -5 - (-2) \cdot 3) + 4(5 \cdot 2 - 0 \cdot 3)\]Simplifying, we have:\[\det(A) = 4(0 + 4) - (-3)(-25 - (-6)) + 4(10)\]\[= 16 - 3 \cdot 19 + 40 = 16 - 57 + 40 = -1\]

Step 3: Construct Matrix A_x and Calculate its Determinant

Replace the first column of \(A\) with \(B\) to get \(A_x\):\[A_x = \begin{bmatrix}10 & -3 & 4 \-2 & 0 & -2 \-9 & 2 & -5\end{bmatrix}\]Calculate \(\det(A_x)\):\[\det(A_x) = 10(0 \cdot -5 - (-2) \cdot 2) - (-3)(-2 \cdot -5 - (-2) \cdot -9) + 4(-2 \cdot 2 - 0 \cdot -9)\]\[= 10(4) - 3(10 + 18) + 4(-4)\]\[= 40 - 84 - 16 = -60\]

Step 4: Construct Matrix A_y and Calculate its Determinant

Replace the second column of \(A\) with \(B\) to get \(A_y\):\[A_y = \begin{bmatrix}4 & 10 & 4 \5 & -2 & -2 \3 & -9 & -5\end{bmatrix}\]Calculate \(\det(A_y)\):\[\det(A_y) = 4(-2 \cdot -5 - (-2) \cdot -9) - 10(5 \cdot -5 - (-2) \cdot 3) + 4(5 \cdot -9 - (-2) \cdot 3)\]\[= 4(10 - 18) - 10(-25 - 6) + 4(-45 + 6)\]\[= 4(-8) + 310 - 156 = -32 + 310 - 156 = 122\]

Step 5: Construct Matrix A_z and Calculate its Determinant

Replace the third column of \(A\) with \(B\) to get \(A_z\):\[A_z = \begin{bmatrix}4 & -3 & 10 \5 & 0 & -2 \3 & 2 & -9\end{bmatrix}\]Calculate \(\det(A_z)\):\[\det(A_z) = 4(0 \cdot -9 - 2 \cdot -2) - (-3)(5 \cdot -9 - 2 \cdot 3) + 10(5 \cdot 2 - 0 \cdot 3)\]\[= 4(0 + 4) + 3(-45 - 6) + 10(10)\]\[= 16 + 153 + 100 = 269\]

Step 6: Use Cramer's Rule to Solve for Variables

Cramer's Rule states that \(x = \frac{\det(A_x)}{\det(A)}\), \(y = \frac{\det(A_y)}{\det(A)}\), \(z = \frac{\det(A_z)}{\det(A)}\).Substitute the determinants:\[x = \frac{-60}{-1} = 60\]\[y = \frac{122}{-1} = -122\]\[z = \frac{269}{-1} = -269\]

Key Concepts

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. These equations are important in many fields such as mathematics, engineering, and economics. In the example provided, we have a system of three equations with three variables: \( x \), \( y \), and \( z \). The goal is to find a set of values for \( x \), \( y \), and \( z \) that satisfy all the equations simultaneously.

Generally, systems can be classified as:

• Consistent: having a unique solution or infinitely many solutions.
• Inconsistent: having no solutions at all.

Solving such systems can be done through various methods, one of which is Cramer’s Rule, particularly useful for small systems with a number of equations equal to the number of unknowns. Cramer's Rule provides an elegant solution using determinants, and is applicable to systems where the determinant of the coefficient matrix is non-zero.

Determinant

The determinant is a special number calculated from a square matrix. It is a crucial component in matrix algebra for solving linear systems, especially when using Cramer’s Rule. In the context of the system of equations, the determinant helps to determine if a unique solution exists.

For a 3x3 matrix, the determinant can be calculated using a specific formula that involves the elements of the matrix. For instance, the determinant of matrix \( A \) in our solution is calculated as:
\[\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]
where \( a, b, c, \ldots \) are elements of the matrix. This determinant is then used in Cramer’s Rule to find the values of the variables. If the determinant is zero, the system may be dependent or inconsistent, and Cramer's Rule cannot be applied.

Matrix Algebra

Matrix algebra involves operations such as addition, subtraction, and multiplication of matrices. For solving systems of linear equations, matrices offer a concise way to represent and manipulate multiple equations simultaneously. In the given problem, the system is represented as:
\[A \cdot x = B\]
where \( A \) is a matrix of coefficients, \( x \) is a column matrix (vector) of variables, and \( B \) is a column matrix of constants. The manipulation of matrices, such as row operations, finding inverses, and calculating determinants, are key operations in matrix algebra. Using Cramer's Rule involves creating matrices \( A_x, A_y, A_z \) by substituting columns with the constants matrix \( B \) and calculating their determinants to find the solution.

Solving Equations

Solving equations, especially linear systems, often involves finding values of variables that make all equations true at the same time. Cramer's Rule is a powerful tool in this regard because it provides a direct formula for the solution if the system is cast into a matrix form and the determinant is non-zero.
Cramer’s Rule uses the formula:

• \(x = \frac{\det(A_x)}{\det(A)}\)
• \(y = \frac{\det(A_y)}{\det(A)}\)
• \(z = \frac{\det(A_z)}{\det(A)}\)

where \( A_x, A_y, A_z \) are matrices formed by replacing corresponding columns in the coefficient matrix \( A \) with the constants vector \( B \). This method ensures that if there's a unique solution, it can be efficiently found and verified. Knowing the solution helps in confirming the consistency of the original system.