Problem 45 - Algebra and Trigonometry

Question

Use Cramer’s Rule to solve the system. $$ \left\\{\begin{aligned} 3 y+5 z &=4 \\ 2 x &-z=10 \\ 4 x+7 y &=0 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{320}{91}, y = \frac{28}{13}, z = -\frac{64}{91} \).

Step 1: Write the System in Matrix Form

First, express the given system of equations in the form of a matrix equation \( AX = B \), where \( A \) is the coefficient matrix, \( X \) is the variable matrix, and \( B \) is the constant matrix. This system is: \( A = \begin{bmatrix} 0 & 3 & 5 \ 2 & 0 & -1 \ 4 & 7 & 0 \end{bmatrix} \), \( X = \begin{bmatrix} x \ y \ z \end{bmatrix} \), \( B = \begin{bmatrix} 4 \ 10 \ 0 \end{bmatrix} \).

Step 2: Calculate the Determinant of A

Find the determinant of matrix \( A \). The determinant \( \Delta \) of \( A \) is calculated as:\[ \Delta = \begin{vmatrix} 0 & 3 & 5 \ 2 & 0 & -1 \ 4 & 7 & 0 \end{vmatrix} \]Applying cofactor expansion along the first row, we get: \[ \Delta = 0 \cdot \begin{vmatrix} 0 & -1 \ 7 & 0 \end{vmatrix} - 3 \cdot \begin{vmatrix} 2 & -1 \ 4 & 0 \end{vmatrix} + 5 \cdot \begin{vmatrix} 2 & 0 \ 4 & 7 \end{vmatrix} \]Calculating the minors and simplifying, we find \( \Delta = -3 (7) + 5 (-14) = -21 - 70 = -91 \).

Step 3: Replace Columns to Find Each Variable

Use Cramer's Rule, which involves replacing the corresponding column in \( A \) with \( B \) and finding determinants.For \( x \), replace the first column of \( A \): \[ A_x = \begin{bmatrix} 4 & 3 & 5 \ 10 & 0 & -1 \ 0 & 7 & 0 \end{bmatrix} \]Calculate determinant \( \Delta_x = \begin{vmatrix} 4 & 3 & 5 \ 10 & 0 & -1 \ 0 & 7 & 0 \end{vmatrix} = 0 (7) + 3 (10) - 5 (70) = 0 + 30 - 350 = -320 \).For \( y \), replace the second column:\[ A_y = \begin{bmatrix} 0 & 4 & 5 \ 2 & 10 & -1 \ 4 & 0 & 0 \end{bmatrix} \]Calculate determinant \( \Delta_y = \begin{vmatrix} 0 & 4 & 5 \ 2 & 10 & -1 \ 4 & 0 & 0 \end{vmatrix} = (0) - 4 (-1) - 5 (40) = 4 - 200 = -196 \).For \( z \), replace the third column:\[ A_z = \begin{bmatrix} 0 & 3 & 4 \ 2 & 0 & 10 \ 4 & 7 & 0 \end{bmatrix}\]Calculate determinant \( \Delta_z = \begin{vmatrix} 0 & 3 & 4 \ 2 & 0 & 10 \ 4 & 7 & 0 \end{vmatrix} = 0 + 3 (40) - 4 (14) = 120 - 56 = 64 \).

Step 4: Apply Cramer's Formula

Using Cramer's Rule, solve for each variable:\( x = \frac{\Delta_x}{\Delta} = \frac{-320}{-91} = \frac{320}{91} \)\( y = \frac{\Delta_y}{\Delta} = \frac{-196}{-91} = \frac{196}{91} = \frac{28}{13} \)\( z = \frac{\Delta_z}{\Delta} = \frac{64}{-91} = -\frac{64}{91} \).

Key Concepts

Matrix Determinant

The concept of a matrix determinant is central when employing methods like Cramer's Rule to solve systems of equations. Essentially, a determinant is a scalar value that can be computed from a square matrix, providing insights into the matrix's properties. In simple terms, it helps determine if a system of equations has a unique solution, and it plays a crucial role in calculating solutions when using matrix-based methods.
When calculating the determinant of a matrix, especially a 3x3 matrix as in this exercise, we utilize cofactor expansion. This process involves expanding the matrix along a row or column, choosing elements that are easy to handle (often those that include zeros to simplify calculations).

• The expression for determinant includes the sign (+/-) alternation when expanding across a row or column.
• Determinants are especially useful because if the determinant is zero, the matrix does not have an inverse, typically indicating infinite or no solutions for the system.

In our exercise, we found the determinant of matrix \( A \) was \(-91\), which indicates that the matrix indeed has an inverse and that each variable can be determined uniquely.

System of Equations

Understanding a system of equations is fundamental to solving them using techniques such as Cramer's Rule. A system of equations is a set of equations with multiple variables that we aim to solve simultaneously. These are prevalent in linear algebra and applied when we want several relationships to hold true simultaneously. Systems can be represented in a variety of ways like in matrix notation

• Each equation corresponds to a row in the matrix form.
• The coefficient matrix (\( A \)) is populated with the coefficients of each variable from each equation.
• Much like puzzles, these systems require strategy and methods for solving.

Handling different situations such as the number of equations equaling the number of unknowns often makes solving systems intriguing yet systematic. Our system involved three equations with three unknowns and was tackled efficiently by writing in matrix form.

Linear Algebra

Linear algebra provides tools and techniques that are essential in understanding and solving problems involving vectors and matrices. It acts as the theoretical foundation for a wide array of fields, especially when dealing with linear equations or transformations. In our context, linear algebra forms the bedrock for comprehending Cramer's Rule which is a straightforward method for solving linear systems using determinants.
Key aspects include:

• The use of matrices and vectors as mathematical tools to simplify complex problems.
• Matrix properties like invertibility, determinants, and eigenvalues explaining deeper system characteristics.
• Enables the use of methods like Cramer's Rule, inversion, or Gaussian elimination.

The ability to manipulate matrices and vector spaces provides a powerful means by which many real-world problems can be represented and solved.

Matrix Equations

Matrix equations are a streamlined way of representing systems of linear equations. Instead of writing multiple equations, we use matrix notation which summarizes a system as a single equation. This concise representation makes the process of manipulating the system, such as using solving techniques like Cramer's Rule, more efficient.
In a matrix equation, we typically have:

• \( A \): the coefficient matrix representing the coefficients of the variables.
• \( X \): the matrix of variables we seek to solve.
• \( B \): the constant matrix representing constant terms from each equation.

Expressing the original equations as a matrix equation \( AX = B \) simplifies calculation, as it prepares the system for matrix manipulation techniques. It's like converting words to a universally understood mathematical language, allowing us to take advantage of powerful tools like determinants and inverses to solve the system effectively.