Problem 12 - Algebra for College Students

Question

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{l}6 x-5 y=1 \\ 4 x-7 y=2\end{array}\right)\)

Step-by-Step Solution

Answer

The solution set is \(x = -\frac{3}{22}\) and \(y = -\frac{4}{11}\).

Step 1: Write the System in Matrix Form

The given system of equations can be represented in matrix form as \(AX = B\), where \(A\) is the coefficient matrix, \(X\) is the variable matrix, and \(B\) is the constant matrix.\[A = \begin{bmatrix} 6 & -5 \ 4 & -7 \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \end{bmatrix}, \quad B = \begin{bmatrix} 1 \ 2 \end{bmatrix}\]

Step 2: Calculate the Determinant of Coefficient Matrix A

The determinant of matrix \(A\), denoted as \(\det(A)\), is calculated as follows:\[\det(A) = \begin{vmatrix} 6 & -5 \ 4 & -7 \end{vmatrix} = (6)(-7) - (4)(-5) = -42 + 20 = -22\]Since the determinant is non-zero, the system has a unique solution.

Step 3: Calculate Determinant of Matrix A_x

Matrix \(A_x\) is formed by replacing the first column of \(A\) with matrix \(B\):\[A_x = \begin{bmatrix} 1 & -5 \ 2 & -7 \end{bmatrix}\]Calculate its determinant:\[\det(A_x) = \begin{vmatrix} 1 & -5 \ 2 & -7 \end{vmatrix} = (1)(-7) - (2)(-5) = -7 + 10 = 3\]

Step 4: Calculate Determinant of Matrix A_y

Matrix \(A_y\) is created by replacing the second column of \(A\) with matrix \(B\):\[A_y = \begin{bmatrix} 6 & 1 \ 4 & 2 \end{bmatrix}\]Calculate its determinant:\[\det(A_y) = \begin{vmatrix} 6 & 1 \ 4 & 2 \end{vmatrix} = (6)(2) - (1)(4) = 12 - 4 = 8\]

Step 5: Apply Cramer's Rule

Cramer's rule states that each variable can be found using the determinants:\[x = \frac{\det(A_x)}{\det(A)} = \frac{3}{-22}, \quad y = \frac{\det(A_y)}{\det(A)} = \frac{8}{-22}\]Thus, we simplify the fractions:\[x = -\frac{3}{22}, \quad y = -\frac{4}{11}\]

Step 6: Conclusion

The solution to the system using Cramer's Rule is \(x = -\frac{3}{22}\) and \(y = -\frac{4}{11}\).

Key Concepts

Determinants

To grasp Cramer's Rule, it is essential to understand the concept of determinants. A determinant is a special number that can be calculated from a matrix. For a 2x2 matrix, the determinant can be easily computed using the formula: \[\text{det}(A) = ad - bc\]This formula refers to a matrix \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\],where you multiply diagonally, subtracting one product from the other.Determinants have some important properties worth noting:

• They help determine whether a set of linear equations, represented by matrices, has a unique solution.
• A non-zero determinant means there's a unique solution, while a zero determinant often indicates infinite solutions or no solution at all.

In our exercise, calculating the determinant was a crucial step. It confirmed that the system of equations has a unique solution.

System of Linear Equations

A system of linear equations involves multiple equations with shared variables. Each equation typically represents a straight line, and solving the system means finding the point(s) where these lines intersect. For a system with two equations like in our original exercise, the goal is to find values for \(x\) and \(y\) that satisfy both equations simultaneously.Why do we use systems of equations?

• They help solve real-world problems involving relationships between different quantities.
• They are fundamental in fields like physics, engineering, and economics.

In the given analytical method, Cramer's Rule provides a straightforward way to find solutions if the conditions are right. These systems can be visualized on a graph, making it easier to understand the solution as an intersection of lines.
Cramer's Rule is particularly useful when dealing with simpler systems where matrices are small, such as in our exercise with 2x2 matrices.

Matrix Algebra

Matrix algebra forms the backbone of many advanced mathematical concepts, including solving systems of linear equations using methods like Cramer's Rule. In matrix algebra, we deal with matrices - rectangular arrays of numbers - which can represent complex data and solve systems in a compact form. Important aspects of matrix algebra include:

• Using matrices to represent systems of equations, which simplifies computation and analysis.
• Performing operations like addition, subtraction, multiplication, and finding determinants.

In our problem, the matrix form of the system helped in organizing coefficients and constants efficiently into the matrices \(A\), \(X\), and \(B\). This simplification allows for the application of Cramer's Rule.
Matrix algebra not only supports problem-solving in theoretical mathematics but also provides tools for disciplines like computer science and statistics. It is vital for anyone thinking deeply about multiple variable problems and data analysis.