Problem 31 - College Algebra

Question

Use Cramer's Rule to solve each system. $$\left\\{\begin{array}{c}4 x-5 y-6 z=-1 \\\x-2 y-5 z=-12 \\\2 x-y\quad\quad=7\end{array}\right.$$

Step-by-Step Solution

Answer

The solutions are \(x=6.16\), \(y=-5\), and \(z=1\)

Step 1: Write down the system matrix, \(A\)

The system matrix \(A\) is formed by taking the coefficients of the variables in the system of equations. Here, that gives us \(\begin{pmatrix} 4 & -5 & -6 \\ 1 & -2 & -5 \\ 2 & -1 & 0 \end{pmatrix}\)

Step 2: Compute the determinant of \(A\)

The determinant, \(|A|\), is a special number that is computed from a matrix. The determinant of a 3x3 can be calculated as \[|A| = a(ei−fh) - b(di−fg) + c(dh−eg)\] Plugging in the values from \(A\) gives: \(|A| = 4((-2) * 0 - (-1) * -5)- (-5)(1*0 - 2 *-5) - 6(1*-1 - 2*2) = -100.\

Step 3: Constructing matrix \(B\), \(C\), and \(D\)

We replace the first, second, and third columns respectively of the system matrix \(A\) with the solution column vector(-1,-12,7), to get three new 3x3 matrices: \(B\), \(C\), and \(D\). Hence, we get: \(B = \begin{pmatrix} -1 & -5 & -6 \\ -12 & -2 & -5 \\ 7 & -1 & 0 \end{pmatrix}\), \(C = \begin{pmatrix} 4 & -1 & -6 \\ 1 & -12 & -5 \\ 2 & 7 & 0 \end{pmatrix}\), and \(D = \begin{pmatrix} 4 & -5 & -1 \\ 1 & -2 & -12 \\ 2 & -1 & 7 \end{pmatrix}\)

Step 4: Compute the determinants of \(B\), \(C\), and \(D\)

Following Step 2 instructions gives: \(|B|=-616\), \(|C|=500\) and \(|D|-100\)

Step 5: Apply Cramer's Rule

Remember that Cramer's Rule says that given a system of equations with system matrix \(A\) and solution column vector \(B\), the solutions are given by \[x=|B|/|A|, y=|C|/|A|, z=|D|/|A|\] Plugging in the computed determinant values, we get: \(x=|B|/|A| = -616/-100 = 6.16\), \(y=|C|/|A| = 500/-100 = -5\), \(z=|D|/|A|=-100/-100=1\)

Key Concepts

Understanding Systems of Linear Equations

A system of linear equations consists of two or more equations that share a set of unknown variables, and the goal is to find the values of these variables. In the exercise, we have a system of three equations with three variables: \( x \), \( y \), and \( z \). To solve such a system, various methods can be applied, including graphing, substitution, elimination, and matrix methods such as Cramer's Rule. Cramer's Rule is a straightforward algorithm that requires the calculation of determinants of matrices, which are arrays of numbers arranged in rows and columns to represent the system's coefficients.

Illustration of the System

In our exercise:

• 4x - 5y - 6z = -1,
• x - 2y - 5z = -12,
• 2x - y = 7,

These equations together form a pattern where the solution satisfies all equations simultaneously. When practicing, it's useful to check that the system is indeed solvable (not inconsistent or dependent), and that the coefficients are correctly aligned when forming matrices.

Determinant Calculation

Determinants are numerical values that are computed from a square matrix. They are crucial in matrix algebra and have various applications, including solving systems of linear equations. The determinant can provide information about the matrix, like whether it's invertible or whether the system of equations has a unique solution. If the determinant of the matrix formed from the coefficients of the equations (the system matrix) is zero, the system has either no solution or infinitely many solutions.

Computing a Determinant

For a 3x3 matrix, the determinant is calculated using a specific formula which involves multiplying and subtracting certain elements of the matrix, as seen in Step 2 of the exercise. Making sure to follow the correct pattern for a 3x3 matrix will help avoid mistakes in calculation.When improving upon existing solutions, it would be beneficial to show the detailed steps of the determinant calculation, thereby reinforcing the concept's understanding.

Matrix Algebra and Cramer's Rule

Matrix algebra is a branch of mathematics that deals with the study of matrices and their operations. It includes addition, subtraction, multiplication, and finding the inverse of matrices, among other things. These operations are the tools that help solve more complex algebraic problems, such as systems of linear equations.Cramer's Rule, an application of matrix algebra, provides a method to solve linear systems when the system matrix is square and its determinant is non-zero. It requires the substitution of each column of the system matrix with the column vector of the solution, to form new matrices. The determinants of these new matrices are then used to find the solutions for the variables.

Applying the Rule

The exercise demonstrates Cramer's Rule by replacing the columns of the original matrix with the constants from the equations, creating new matrices and computing their determinants. The final values of \( x \), \( y \), and \( z \) are found by dividing these determinants by the original matrix's determinant. It is essential to understand matrix algebra to follow the rule's steps and interpret the results correctly.Incorporating visual aids, such as step-by-step determinant calculations or matrix substitutions, could greatly improve comprehension for students who are visual learners.