Problem 24 - Intermediate Algebra

Question

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{rr} 5 x+2 y= & -15 \\ 7 x-3 y= & 37 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution is \( x = 1 \) and \( y = -10 \).

Step 1: Write the system of equations

The given system of equations is: \[ \begin{align*} 5x + 2y &= -15, \ 7x - 3y &= 37. \end{align*} \]

Step 2: Identify the coefficient matrix

The coefficient matrix for the system is given by: \[ A = \begin{bmatrix} 5 & 2 \ 7 & -3 \end{bmatrix}. \]

Step 3: Calculate the determinant of the coefficient matrix

The determinant of the matrix \( A \) is evaluated as follows:\[ \text{det}(A) = \begin{vmatrix} 5 & 2 \ 7 & -3 \end{vmatrix} = (5)(-3) - (2)(7) = -15 - 14 = -29. \]

Step 4: Replace the x-column and calculate the determinant

Replace the first column of matrix \( A \) by the constants from the right side of the equations to form matrix \( A_x \):\[ A_x = \begin{bmatrix} -15 & 2 \ 37 & -3 \end{bmatrix}. \]Calculate the determinant:\[ \text{det}(A_x) = \begin{vmatrix} -15 & 2 \ 37 & -3 \end{vmatrix} = (-15)(-3) - (2)(37) = 45 - 74 = -29. \]

Step 5: Replace the y-column and calculate the determinant

Replace the second column of matrix \( A \) by the constants from the right side of the equations to form matrix \( A_y \):\[ A_y = \begin{bmatrix} 5 & -15 \ 7 & 37 \end{bmatrix}. \]Calculate the determinant:\[ \text{det}(A_y) = \begin{vmatrix} 5 & -15 \ 7 & 37 \end{vmatrix} = (5)(37) - (-15)(7) = 185 + 105 = 290. \]

Step 6: Apply Cramer's Rule to find x and y

Cramer's rule states: \( x = \frac{\text{det}(A_x)}{\text{det}(A)} \) and \( y = \frac{\text{det}(A_y)}{\text{det}(A)} \).For \( x \): \[ x = \frac{-29}{-29} = 1. \]For \( y \): \[ y = \frac{290}{-29} = -10. \]

Step 7: State the solution set

The solution to the system of equations is \( x = 1 \) and \( y = -10 \).

Key Concepts

Determinant of a Matrix

A matrix can be thought of as a rectangular arrangement of numbers, symbols, or expressions in rows and columns. To solve equations using Cramer's Rule, understanding determinants is crucial. The determinant of a matrix provides valuable information about the matrix, allowing us to solve systems of linear equations in a structured way.

For a 2x2 matrix:

• The determinant is calculated as: \( \text{det}(A) = a_{11}a_{22} - a_{12}a_{21} \).
• This formula involves multiplying the top-left and bottom-right element, then subtracting the product of the top-right and bottom-left element.
• The determinant's value indicates if a unique solution exists. If the determinant is zero, the system may have no unique solutions.

Calculating the determinant precisely helps apply Cramer's Rule effectively. It's a stepping stone to solving more complex linear systems.

System of Linear Equations

Systems of linear equations consist of mathematical equations that relate multiple variables. These systems can either have one solution, no solution, or infinitely many solutions. The problem you have tackled is a system of two linear equations with two unknowns, which looks like:

\[\begin{align*}ax + by &= e, \mx + ny &= f.\end{align*}\]

Here:

• \( x \) and \( y \) are variables.
• \( a, b, m, \) and \( n \) are coefficients represented in the matrix.
• \( e \) and \( f \) are constants on the right side of the equations.

To find a solution using methods like substitution, elimination, or through matrix algebra like Cramer's Rule, the goal is to express the unknowns in terms of constants. In this exercise, you solve the system using Cramer's Rule by leveraging determinants of matrices to find precise values of \( x \) and \( y \).

Matrix Algebra

Matrix Algebra is a powerful mathematical tool used to handle linear equations. It uses matrices to represent and solve systems. Understanding basics of matrix operations is key to tackling algebraic problems like the given exercise. Here's how it applies:

• Matrices Representation: Coefficients of linear equations are represented in matrices. For example, a 2x2 matrix holds coefficients of \( x \) and \( y \).
• Matrix Operations: Includes addition, subtraction, multiplication, and calculating determinants. These operations help in manipulating the systems of equations to arrive at solutions.
• Cramer's Rule: Utilizes determinants to solve for variables when the coefficient matrix's determinant is non-zero.

By forming the coefficient matrix \( A \), and replacing columns with constants (to form matrices \( A_x \) and \( A_y \)), you can use determinant operations to find solutions to the system of linear equations efficiently using Cramer's Rule.