Problem 41 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 33-42: Use Cramer's rule, whenever applicable, to solve the system. $$ \left\\{\begin{array}{rr} 5 x+2 y-z= & -7 \\ x-2 y+2 z= & 0 \\ 3 y+z= & 17 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{40}{37}\), \(y = \frac{91}{37}\), \(z = \frac{77}{37}\).

Step 1: Write the System in Matrix Form

Express the given system of equations as a matrix equation \(AX = B\). Here,\[ A = \begin{bmatrix} 5 & 2 & -1 \ 1 & -2 & 2 \ 0 & 3 & 1 \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad B = \begin{bmatrix} -7 \ 0 \ 17 \end{bmatrix} \] The equation can be rewritten as \(AX = B\).

Step 2: Compute the Determinant of A

Calculate the determinant of matrix \(A\). \[\text{det}(A) = \begin{vmatrix} 5 & 2 & -1 \ 1 & -2 & 2 \ 0 & 3 & 1 \end{vmatrix} = 5(-2 \cdot 1 - 3 \cdot 2) - 2(1 \cdot 1 - 2 \cdot 0) - 1(1 \cdot 3 - (-2) \cdot 0) = -37\]

Step 3: Compute the Determinant of Each Modified Matrix

To use Cramer's rule, calculate the determinant for each modified matrix where one column of \(A\) is replaced by \(B\).- For determinant with \(B\) replacing the 1st column:\[ D_x = \begin{vmatrix} -7 & 2 & -1 \ 0 & -2 & 2 \ 17 & 3 & 1 \end{vmatrix} = 40 \]- For determinant with \(B\) replacing the 2nd column:\[ D_y = \begin{vmatrix} 5 & -7 & -1 \ 1 & 0 & 2 \ 0 & 17 & 1 \end{vmatrix} = -91 \]- For determinant with \(B\) replacing the 3rd column:\[ D_z = \begin{vmatrix} 5 & 2 & -7 \ 1 & -2 & 0 \ 0 & 3 & 17 \end{vmatrix} = -77 \]

Step 4: Solve for Each Variable Using Cramer's Rule

Use Cramer's rule to find each variable:- \(x = \frac{D_x}{\text{det}(A)} = \frac{40}{-37} = -\frac{40}{37}\)- \(y = \frac{D_y}{\text{det}(A)} = \frac{-91}{-37} = \frac{91}{37}\)- \(z = \frac{D_z}{\text{det}(A)} = \frac{-77}{-37} = \frac{77}{37}\)

Key Concepts

Matrix Determinants

Matrix determinants are special numbers that can be calculated from a square matrix. They play a crucial role in various areas of algebra, especially in solving systems of linear equations using methods like Cramer's Rule. For a 3x3 matrix, the determinant can be computed using the formula:

• Multiply the diagonal entries in several ways and sum the results, while subtracting some products according to specified rules.
• The determinant helps determine whether a matrix is singular or non-singular.

This specific computation is pivotal. If the determinant is zero, the matrix is singular, meaning it doesn't have an inverse and the system of equations has no unique solution. In our example, we calculated the determinant of matrix \(A\) as -37, which indicates that it's non-singular. This allowed us to proceed with solving the system using Cramer's Rule.

Systems of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. Each equation is a line, and the solution to the system is where these lines intersect. There are several methods to solve such systems, but they primarily hinge on understanding how matrices can represent them.

• In a system like ours, each equation can be converted into a row of a matrix.
• The system is usually represented in the form \(AX = B\), where \(A\) is the coefficient matrix, \(X\) is the variable matrix, and \(B\) is the constant terms matrix.

Understanding systems of linear equations is essential, as they appear in many scientific and engineering problems. Solving these systems can determine correlations and predict outcomes.

Matrix Algebra

Matrix algebra is a fundamental compute framework for dealing with equations involving matrices. It includes operations like addition, subtraction, and multiplication. One central idea is transforming systems of linear equations into matrix equations, which simplifies the process of finding solutions.

• Matrix multiplication isn't straightforward; it involves the dot product of rows and columns.
• In equations, understanding matrix multiplication lets us efficiently solve complex systems and explore various transformations.

Matrix algebra simplifies many concepts in algebra by providing a structured way of computing and can solve multiple linear equations at once. In our exercise, it made rewriting and manipulating these equations into \(AX = B\) possible, opening the path for techniques like Cramer's Rule.

Solving Equations

Solving equations can involve several methods depending on the nature of the equations. For systems of linear equations, methods include substitution, elimination, and using determinants with Cramer's Rule. Each method has its own application and efficiency:

• Cramer's Rule is efficient for systems where the number of equations equals the number of unknowns.
• With non-zero determinants, Cramer's Rule computes solutions by substituting different columns of the coefficient matrix with the constant matrix \(B\) and finding determinants.

This rule is particularly useful because it provides a direct way to solve for each variable independently using the determinants we discussed. It ensures exact solutions in systems where applicable. As seen in our exercise, we calculated determinants for each variable, using them to find exact solutions for \(x\), \(y\), and \(z\).