Problem 45 - Intermediate Algebra

Question

Use Cramer's rule to solve each system of equations. $$ \left\\{\begin{array}{l} 2 x+3 y=31 \\ 3 x+2 y=39 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \( x = 11 \) and \( y = 3 \).

Step 1: Define the Determinant of the Coefficient Matrix

To use Cramer's rule, first form the coefficient matrix based on the system of equations. The matrix is \( A = \begin{bmatrix} 2 & 3 \ 3 & 2 \end{bmatrix} \). The determinant of this matrix \( A \) is calculated as follows: \( \text{det}(A) = (2)(2) - (3)(3) = 4 - 9 = -5 \).

Step 2: Calculate Determinant for Variable x

To find \( x \), replace the first column of matrix \( A \) with the constant terms from the equations to form matrix \( A_x = \begin{bmatrix} 31 & 3 \ 39 & 2 \end{bmatrix} \). Calculate its determinant: \( \text{det}(A_x) = (31)(2) - (39)(3) = 62 - 117 = -55 \).

Step 3: Calculate Determinant for Variable y

To find \( y \), replace the second column of matrix \( A \) with the constant terms to form matrix \( A_y = \begin{bmatrix} 2 & 31 \ 3 & 39 \end{bmatrix} \). Calculate its determinant: \( \text{det}(A_y) = (2)(39) - (3)(31) = 78 - 93 = -15 \).

Step 4: Solve for Variables Using Determinants

Use Cramer's rule to solve for \( x \) and \( y \). The formula for \( x \) is \( x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{-55}{-5} = 11 \). The formula for \( y \) is \( y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{-15}{-5} = 3 \).

Key Concepts

Understanding a System of Equations

A system of equations is a set of two or more equations with the same set of variables. In this exercise, we are dealing with two equations and two variables, both being linear. The system of equations can be written in the form:

• Equation 1: \(2x + 3y = 31\)
• Equation 2: \(3x + 2y = 39\)

The goal is to find values of \(x\) and \(y\) that satisfy both equations simultaneously. Systems of equations are useful because they represent real-world situations where multiple conditions must be met. By solving these systems, we can find the unique solution that often represents the intersection point on a graph where both lines meet. Such graphs help in visualizing solutions, especially in simple linear systems like this one. Bridging this understanding through algebraic methods allows students to solve more complex systems.

Calculating the Determinant of a Matrix

The determinant of a matrix serves as a key player in solving systems of equations, especially when employing Cramer's Rule. A square matrix, like our 2x2 coefficient matrix, has a determinant which can be calculated with a straightforward formula. For a matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is \(\text{det}(A) = ad - bc\). This exercise involves the coefficient matrix:

• \(A = \begin{bmatrix} 2 & 3 \ 3 & 2 \end{bmatrix}\)
  \(\text{det}(A) = (2)(2) - (3)(3) = 4 - 9 = -5\)

The determinant tells us about the system's properties. If it is zero, the system may have no solution or infinitely many solutions, meaning the equations are dependent or parallel. A non-zero determinant, like in our example (\(-5\)), indicates a unique solution, making the system consistent and independent.

Solving Linear Equations Using Cramer's Rule

To solve linear equations using Cramer's Rule, we need to compute determinants for matrices derived from the original coefficient matrix by replacing columns with the constants from the equations. This helps isolate each variable one at a time:

• For \( x \), replace the first column of the coefficient matrix with the constants: \(A_x = \begin{bmatrix} 31 & 3 \ 39 & 2 \end{bmatrix}\)
  \(\text{det}(A_x) = (31)(2) - (39)(3) = -55\)
• Solve for \(x\) using: \( x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{-55}{-5} = 11 \)
• For \( y \), replace the second column with the constants: \(A_y = \begin{bmatrix} 2 & 31 \ 3 & 39 \end{bmatrix}\)
  \(\text{det}(A_y) = (2)(39) - (3)(31) = -15\)
• Solve for \( y \) using: \( y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{-15}{-5} = 3 \)

The values \(x = 11\) and \(y = 3\) are thus the solutions to the system that satisfies both equations. Cramer's Rule provides a straightforward method for solving such linear systems, especially when the number of equations and unknowns match.