Problem 19 - Precalculus with Limits

Question

In Exercises 17-20, use a graphing utility and Cramer's Ruleto solve (if possible) the system of equations. \(\begin{cases} 2x - y + z = 5 \\ x - 2y - z = 1 \\ 3x + y + z = 4 \end{cases}\)

Step-by-Step Solution

Answer

The solution for the system of equations using Cramer's Rule will be the values of x, y, z obtained in Step 5.

Step 1: Calculation of Determinant

Firstly, prepare a coefficient matrix A from given equations as: \[A = \begin{bmatrix} 2 & -1 & 1 \ 1 & -2 & -1 \ 3 & 1 & 1 \end{bmatrix}\] Then, calculate its determinant |\(A\)|.

Step 2: Calculation of Dx

Now, replace the x column in the matrix A with constants to get Dx as: \[Dx = \begin{bmatrix} 5 & -1 & 1 \ 1 & -2 & -1 \ 4 & 1 & 1 \end{bmatrix}\] Then, calculate the determinant |\(Dx\)|.

Step 3: Calculation of Dy

Replace the y column in matrix A with constants to get Dy as: \[Dy = \begin{bmatrix} 2 & 5 & 1 \ 1 & 1 & -1 \ 3 & 4 & 1 \end{bmatrix}\] Then, calculate the determinant |\(Dy\)|.

Step 4: Calculation of Dz

Replace the z column in matrix A with constants to get Dz as: \[Dz = \begin{bmatrix} 2 & -1 & 5 \ 1 & -2 & 1 \ 3 & 1 & 4 \end{bmatrix}\] Then, calculate the determinant |\(Dz\)|.

Step 5: Solving for Variables

Now, solve for the variables using Cramer's Rule, which states: \(x = \frac{|Dx|}{|A|}\), \(y = \frac{|Dy|}{|A|}\), \(z = \frac{|Dz|}{|A|}\)

Key Concepts

Determinant

The determinant is a special number that you can calculate from a square matrix. Think of it as a value that helps you understand characteristics of the matrix, such as whether it is invertible or not. In the context of Cramer's Rule, the determinant of the coefficient matrix is crucial.
When dealing with a matrix like \[A = \begin{bmatrix} 2 & -1 & 1 \ 1 & -2 & -1 \ 3 & 1 & 1 \end{bmatrix}\]we find the determinant, denoted \(|A|\), by performing operations on the elements of the matrix. This involves multiplying and adding the elements in a specific fashion.
To calculate the determinant of a 3x3 matrix, you follow a specific pattern involving permutations of its elements. This calculation determines if the system of equations has a unique solution (i.e., if the determinant is non-zero) or if it’s dependent or inconsistent (i.e., if the determinant is zero).
Understanding determinants is fundamental in linear algebra, especially when solving systems of linear equations using Cramer's Rule.

System of Equations

A system of equations refers to multiple equations that are solved simultaneously because they share variables. In these equations, each line or equation typically represents a different piece of information about the same variables.
For example, in the system given:\[\begin{cases} 2x - y + z = 5 \ x - 2y - z = 1 \ 3x + y + z = 4 \end{cases}\]- All three equations must be true at the same time.
- The values of \(x\), \(y\), and \(z\) are what you're trying to find.

Solving such a system involves finding those values for the variables that make all the equations hold true. Cramer's Rule is one method for solving such systems, particularly when the equations are expressed in terms of a coefficient matrix. A key part of this solution method involves ensuring that the determinant of the coefficient matrix is non-zero, which guarantees a unique solution.

Coefficient Matrix

A coefficient matrix is a way of organizing the coefficients of a system of linear equations into a matrix form. This matrix simplifies the process of solving the system, especially when using methods like Cramer's Rule.
For the system:\[\begin{cases} 2x - y + z = 5 \ x - 2y - z = 1 \ 3x + y + z = 4 \end{cases}\]we extract the coefficients of the variables to form the coefficient matrix:\[A = \begin{bmatrix} 2 & -1 & 1 \ 1 & -2 & -1 \ 3 & 1 & 1 \end{bmatrix}\]The rows of this matrix correspond to the equations, and the columns correspond to the variables. This matrix serves as the foundation for constructing other matrices like \(D_x, D_y, D_z\) (used in Cramer's Rule) by replacing specific columns with the constant terms from the equations.
A firm grasp of the structure and role of the coefficient matrix is essential. It transforms complex systems of equations into manageable matrix operations, paving the way for systematic solutions.