Problem 31 - Algebra and Trigonometry

Question

\(29-44\) Use Cramer's Rule to solve the system. $$ \left\\{\begin{array}{c}{x-6 y=3} \\ {3 x+2 y=1}\end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{3}{5} \), \( y = -\frac{2}{5} \).

Step 1: Identifying Coefficient Matrix

First, identify the coefficient matrix of the system of equations. The coefficient matrix is \( A = \begin{pmatrix} 1 & -6 \ 3 & 2 \end{pmatrix} \).

Step 2: Calculating Determinant of Coefficient Matrix

Calculate the determinant of the coefficient matrix \( A \), denoted as \( \det(A) \). For a 2x2 matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is given by \( ad - bc \). Thus, \( \det(A) = (1)(2) - (-6)(3) = 2 + 18 = 20 \).

Step 3: Forming Matrices for Numerators

Create matrices \( A_x \) and \( A_y \) by replacing the respective columns of \( A \) with the constants from the right-hand side of the equations. For \( A_x = \begin{pmatrix} 3 & -6 \ 1 & 2 \end{pmatrix} \) and for \( A_y = \begin{pmatrix} 1 & 3 \ 3 & 1 \end{pmatrix} \).

Step 4: Calculating Determinant for Numerator of x

Calculate \( \det(A_x) \). For \( A_x = \begin{pmatrix} 3 & -6 \ 1 & 2 \end{pmatrix} \), \( \det(A_x) = (3)(2) - (-6)(1) = 6 + 6 = 12 \).

Step 5: Calculating Determinant for Numerator of y

Calculate \( \det(A_y) \). For \( A_y = \begin{pmatrix} 1 & 3 \ 3 & 1 \end{pmatrix} \), \( \det(A_y) = (1)(1) - (3)(3) = 1 - 9 = -8 \).

Step 6: Applying Cramer's Rule

According to Cramer's Rule, \( x = \frac{\det(A_x)}{\det(A)} \) and \( y = \frac{\det(A_y)}{\det(A)} \). Substitute the determinants: \( x = \frac{12}{20} = \frac{3}{5} \) and \( y = \frac{-8}{20} = -\frac{2}{5} \).

Step 7: Conclusion

The solution to the system of equations using Cramer's Rule is \( x = \frac{3}{5} \) and \( y = -\frac{2}{5} \).

Key Concepts

System of Equations

A system of equations is a set of two or more equations with the same variables. Solving such systems means finding values for the variables that satisfy all the equations simultaneously.
This can be imagined as finding a point where all the equations, represented as lines (or planes in higher dimensions), intersect.
In algebra, various methods exist for solving these systems, each with its benefits and limitations. One common method is substitution, where you solve one equation for one variable and substitute it back into the others. Another is elimination, where you add or subtract equations to eliminate a variable, simplifying the system.
However, Cramer's Rule offers a more systematic approach, especially suitable for systems where the number of equations matches the number of variables and when we can form square matrices from the coefficients:

• Advantages include direct computation for both variables.
• Each solution involves determinants, frequently giving insight into whether a unique solution exists, or if parallel lines (no solution) or infinitely many solutions (same line) occur.

Determinants

Determinants are mathematical values that can be calculated from square matrices. They offer insights into the system of equations that the matrix represents.
For a 2x2 matrix, the determinant can easily be found using the formula \( ad-bc \), where \( a, b, c, \) and \( d \) are elements of the matrix. This is simple yet powerful because the determinant indicates several key properties of the matrix:

• A non-zero determinant signifies that the matrix (and hence the corresponding system of equations) has a unique solution.
• A zero determinant, however, indicates the system may have no solutions or infinitely many solutions, usually associated with parallel or identical lines in 2D.

The determinant is central in Cramer's Rule, where it helps find values of variables directly by setting up different matrices and comparing them to the original coefficient matrix.For instance, determinants are calculated for the numerator matrices \( A_x \) and \( A_y \), which are derived by replacing either column of the coefficient matrix with the constants from the system of equations.
This unique property ties the coefficients directly to the solution of the system.

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with matrices and the operations that can be performed on them. It involves understanding structures comprised of rows and columns, emphasizing how these can be used to solve systems of equations.
Matrix algebra simplifies many problems and processes, especially those involving linear equations. At its core, a matrix is a rectangular array of numbers that can represent a system of equations. Each matrix in an equation system encapsulates coefficients, constants, and offers the main toolset for methods like Cramer's Rule:

• Square matrices, where rows equal columns in number, allow computation of determinants, a crucial step in such methods.
• Matrix multiplication and inversion are other significant operations, though not directly used in Cramer's Rule, they underline the concept's importance.

Using a matrix to represent equations helps understand the relationships among variables and ultimately permits advanced operations to derive solutions efficiently. Learning matrix algebra deeply enriches your mathematical toolbox, aligning practical computation with theoretical insights.