Problem 32 - College Algebra

Question

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{r} 4 x-5 y=7 \\ -3 x+9 y=0 \end{array} $$

Step-by-Step Solution

Answer

The solution to the system is \( x = 3 \) and \( y = 1 \).

Step 1: Write Down the System in Matrix Form

Let's express the given system of equations in matrix form. The system is: \( 4x - 5y = 7 \) \( -3x + 9y = 0 \) In matrix form, this system is represented as \( AX = B \), where: \( A = \begin{bmatrix} 4 & -5 \ -3 & 9 \end{bmatrix} \) \( X = \begin{bmatrix} x \ y \end{bmatrix} \) \( B = \begin{bmatrix} 7 \ 0 \end{bmatrix} \).

Step 2: Calculate the Determinant of Matrix A

The determinant of matrix \( A \) can be found using the formula for a 2x2 matrix, \( \text{det}(A) = a_{11}a_{22} - a_{12}a_{21} \). So, \( \text{det}(A) = (4)(9) - (-5)(-3) = 36 - 15 = 21 \).

Step 3: Find Determinant of A_x

Matrix \( A_x \) is obtained by replacing the first column of \( A \) with vector \( B \): \( A_x = \begin{bmatrix} 7 & -5 \ 0 & 9 \end{bmatrix} \). The determinant \( \text{det}(A_x) = 7 \times 9 - (-5) \times 0 = 63 \).

Step 4: Find Determinant of A_y

Matrix \( A_y \) is formed by replacing the second column of \( A \) with vector \( B \): \( A_y = \begin{bmatrix} 4 & 7 \ -3 & 0 \end{bmatrix} \). The determinant \( \text{det}(A_y) = 4 \times 0 - 7 \times (-3) = 21 \).

Step 5: Apply Cramer's Rule

Using Cramer's Rule: \( x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{63}{21} = 3 \) \( y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{21}{21} = 1 \).

Step 6: Verify the Solution

Substitute \( x = 3 \) and \( y = 1 \) back into the original equations to verify: 1. \( 4(3) - 5(1) = 12 - 5 = 7 \) (True) 2. \( -3(3) + 9(1) = -9 + 9 = 0 \) (True) Both equations hold true, confirming the solution is correct.

Key Concepts

Determinant

The concept of a determinant is crucial when solving systems of linear equations using Cramer's Rule. The determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix, it serves as a measure of the area scaling factor and can determine if the matrix is invertible.
To compute the determinant of a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the formula is \( \text{det}(A) = ad - bc \).
In the provided exercise, the determinant of matrix \( A \) is \( 36 - 15 = 21 \). A non-zero determinant, such as 21, indicates that the system of equations has a unique solution. It is important to compute the determinant correctly, as it directly influences the solution of the system.

System of Linear Equations

A system of linear equations consists of multiple linear equations that share the same set of variables. The goal is to find values for these variables that satisfy all the equations in the system.
In our example, we have the system:

• \( 4x - 5y = 7 \)
• \(-3x + 9y = 0 \)

The solution to a system is a pair of values \((x, y)\) that ensures each equation is true. Systems can be solved in various ways, such as by graphing, substitution, elimination, or using matrices and determinants as demonstrated with Cramer's Rule. Understanding each method broadens your problem-solving toolkit.

Matrix Representation

Matrix representation is a method of organizing a system of linear equations into a compact form. This approach makes the application of Cramer's Rule straightforward and efficient.
A system \( AX = B \) consists of:

• Matrix \( A \), containing the coefficients of the variables.
• Matrix \( X \), a column vector of the variables.
• Matrix \( B \), a column vector with the constants from the right-hand side of the equations.

For our exercise, the matrix representation is:
\( A = \begin{bmatrix} 4 & -5 \ -3 & 9 \end{bmatrix} \), \( X = \begin{bmatrix} x \ y \end{bmatrix} \), and \( B = \begin{bmatrix} 7 \ 0 \end{bmatrix} \).
Using this notation, Cramer's Rule calculates the variable values by finding determinants.

Verification of Solution

After finding a purported solution using Cramer's Rule, it's essential to verify that the solution satisfies the original system of linear equations. This step ensures the accuracy and correctness of the solution.
To verify, substitute the obtained values of \( x = 3 \) and \( y = 1 \) back into the original equations and check:

• For \( 4x - 5y = 7 \): \( 4(3) - 5(1) = 12 - 5 = 7 \), which is true.
• For \( -3x + 9y = 0 \): \( -3(3) + 9(1) = -9 + 9 = 0 \), which is also true.

Both equations hold, confirming that the solution \((x, y) = (3, 1)\) is indeed correct. This verification process is a crucial step in ensuring that the system's solution is reliable.