Problem 25 - College Algebra

Question

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{l} 2 x-3 y=-1 \\ 4 x+5 y=9 \end{array} $$

Step-by-Step Solution

Answer

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

Step 1: Write Down the System of Equations

The given system of equations is:\[ \begin{align*} 2x - 3y &= -1 \ 4x + 5y &= 9 \end{align*} \]

Step 2: Find the Coefficient Matrix

The coefficient matrix \( A \) for the given system of equations is:\[ A = \begin{bmatrix} 2 & -3 \ 4 & 5 \end{bmatrix} \]

Step 3: Calculate the Determinant of the Coefficient Matrix

To find the determinant of matrix \( A \), use the formula:\[ \text{det}(A) = (2)(5) - (-3)(4) = 10 + 12 = 22 \]

Step 4: Set Up Determinants for Numerator Matrices

Substitute the values from the constants on the right hand side of the equations to create matrices for \( x \) and \( y \).Matrix for \( x \):\[ A_x = \begin{bmatrix} -1 & -3 \ 9 & 5 \end{bmatrix} \]Matrix for \( y \):\[ A_y = \begin{bmatrix} 2 & -1 \ 4 & 9 \end{bmatrix} \]

Step 5: Calculate Determinant for Matrix A_x

Calculate the determinant of \( A_x \):\[ \text{det}(A_x) = (-1)(5) - (-3)(9) = -5 + 27 = 22 \]

Step 6: Calculate Determinant for Matrix A_y

Calculate the determinant of \( A_y \):\[ \text{det}(A_y) = (2)(9) - (-1)(4) = 18 + 4 = 22 \]

Step 7: Apply Cramer's Rule to Find x and y

Using Cramer's Rule, the solutions are:\[ x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{22}{22} = 1 \]\[ y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{22}{22} = 1 \]

Step 8: Conclusion

The solution to the system of equations using Cramer's Rule is \( x = 1 \) and \( y = 1 \).

Key Concepts

Linear Equations

Linear equations are a fundamental part of algebra that deal with relationships between variables. In mathematics, a linear equation is any equation that can be written in the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables. These equations represent straight lines when plotted on a graph.
To illustrate, consider the system of linear equations given in the exercise:

• \( 2x - 3y = -1 \)
• \( 4x + 5y = 9 \)

Each equation describes a line, and solving the system means finding the values of \( x \) and \( y \) where these lines intersect. In this case, Cramer's Rule is applied to find this intersection point.

Determinant

The determinant is a special number calculated from a square matrix. It provides important properties of the matrix, such as whether the matrix is invertible, and is crucial in solving systems of linear equations, especially using Cramer's Rule.
For a 2x2 matrix, the determinant can be calculated using the formula:

• \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \)
• \( \text{det}(A) = ad - bc \)

In our example, the determinant of the coefficient matrix \( A \) is calculated as \( 22 \). This positive number indicates that the matrix is invertible, thus confirming the system has a unique solution. The result of the determinant is crucial for Cramer's Rule, as it is used in the denominator for finding the values of \( x \) and \( y \).

Coefficient Matrix

The coefficient matrix is composed of the coefficients of the variables from the given system of linear equations.
For the exercise at hand, the coefficient matrix \( A \) is:

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

This matrix is derived by extracting the numerical coefficients from the equations:

• Equation 1: \( 2x - 3y = -1 \), coefficients \( 2 \) and \( -3 \)
• Equation 2: \( 4x + 5y = 9 \), coefficients \( 4 \) and \( 5 \)

The coefficient matrix is indispensable when using methods like Cramer's Rule because it sets the foundation for calculating the necessary determinants.

System of Equations

A system of equations consists of two or more equations involving the same set of variables. To solve a system of equations means finding the values for the variables that satisfy all equations simultaneously. This is where the methods like substitution, elimination, and Cramer's Rule come into play.
The given problem involves a system of linear equations:

• \( 2x - 3y = -1 \)
• \( 4x + 5y = 9 \)

Such a system can be visualized on a graph, with each equation representing a line. The solution to the system is the point where these lines intersect. Cramer's Rule specifically works for systems where the number of equations equals the number of variables, providing a numerical method to find a unique solution. By using determinants of matrices, this method simplifies the process of finding \( x \) and \( y \). In our problem, both equations and their solution are straightforward, with the intersection point being \( (1, 1) \).