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A system of equations is a collection of two or more equations with the same set of variables. Solving these systems involves finding values for the variables that satisfy all equations simultaneously. For example, the system of equations given in the exercise is represented as:

\( \frac{1}{4} x + \frac{2}{3} y = 25 \)
\( \frac{3}{5} x - \frac{1}{10} y = 12 \).

There are various methods to solve these systems, including substitution, elimination, and matrix methods like Cramer's Rule. Cramer's Rule is particularly useful for solving linear systems with the same number of equations as unknowns.

A determinant is a special number calculated from a square matrix. It provides important properties of the matrix, such as whether the matrix is invertible. For a 2x2 matrix \(A\) represented as \[ A = \begin{bmatrix} a & b \ c & d \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{bmatrix} \]

the determinant is calculated as \( \text{det}(A) = ad - bc \).

In the exercise, the determinant of the coefficient matrix \(A\) is computed as follows:

\( \text{det}(A) = \frac{1}{4} \times (-\frac{1}{10}) - \frac{2}{3} \times \frac{3}{5} \ \)
\( \text{det}(A) = -\frac{1}{40} - \frac{6}{15} \ \)
\( \text{det}(A) = -\frac{1}{40} - \frac{16}{40} \ \)
\( \text{det}(A) = -\frac{17}{40} \).

Matrix algebra involves the manipulation of matrices to solve systems of linear equations and other mathematical problems. In this context, matrices are rectangular arrangements of numbers, symbols, or expressions in rows and columns. Here’s how matrices come into play:

• The coefficient matrix \( A \) for the system is: \[ A = \begin{bmatrix} \frac{1}{4} & \frac{2}{3} \ \frac{3}{5} & -\frac{1}{10} \ \ \ \ \ \end{bmatrix} \]
• The constants matrix \( C \): \[ C = \begin{bmatrix} 25 \ 12 \ \ \ \ \end{bmatrix} \]
• Modified matrices for Cramer's Rule (replacing columns of \( A \) with \( C \)): \[ A_{Dx} = \begin{bmatrix} 25 & \frac{2}{3} \ 12 & -\frac{1}{10} \ \ \ \ \end{bmatrix} \], \[ A_{Dy} = \begin{bmatrix} \frac{1}{4} & 25 \ \frac{3}{5} & 12 \ \ \ \ \end{bmatrix} \]

Matrix algebra simplifies the calculation of determinants, as shown in the solution steps.

Linear algebra is a field of mathematics focusing on vector spaces, and linear transformations between those spaces. It includes concepts like vectors, matrices, and systems of linear equations. These are the fundamental tools used in solving our given system using Cramer's rule.

Linear equations like \( \frac{1}{4} x + \frac{2}{3} y = 25 \) form the basis of systems analyzed in linear algebra.

Cramer's rule specifically leverages matrix algebra and determinants to find the unique solutions of such systems. In this exercise, both concepts ensure a structured approach from formulation to solution:

• Solve for \(x\) using: \( x = \frac{ \text{det}(A_{Dx}) }{ \text{det}(A) } \)
• Solve for \(y\) using: \( y = \frac{ \text{det}(A_{Dy}) }{ \text{det}(A) } \)


Finding values of \(x\) and \(y\) becomes systematic and automated using these rules, highlighting the power of linear algebra in solving real-world problems.