Question

Use Cramer's rule to solve the following equation systems: \((a) 3 x_{1}-2 x_{2}=6\) \(2 x_{1}+x_{2}=11\) \(\langle b\rangle-x_{1}+3 x_{2}=-3\) \(4 x_{1}-x_{2}=12\) (c) \(8 x_{1}-7 x_{2}=9\) \(x_{1}+x_{2}=3\) \((d) 5 x_{1}+9 x_{2}=14\) \(7 x_{1}-3 x_{2}=4\)

Short answer

Use Cramer's Rule to find determinants and solve for each variable in the given systems.

Step-by-step solution

Identify Coefficients and Constants

For each system of equations, identify the coefficients of the variables and the constant terms. We will label these as follows:(a) System:- Coefficients: \( a_{11} = 3, a_{12} = -2, a_{21} = 2, a_{22} = 1 \)- Constants: \( b_{1} = 6, b_{2} = 11 \)(b) System:- Coefficients: \( a_{11} = -1, a_{12} = 3, a_{21} = 4, a_{22} = -1 \)- Constants: \( b_{1} = -3, b_{2} = 12 \)(c) System:- Coefficients: \( a_{11} = 8, a_{12} = -7, a_{21} = 1, a_{22} = 1 \)- Constants: \( b_{1} = 9, b_{2} = 3 \)(d) System:- Coefficients: \( a_{11} = 5, a_{12} = 9, a_{21} = 7, a_{22} = -3 \)- Constants: \( b_{1} = 14, b_{2} = 4 \).

Key concepts

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. The solution to such systems is the set of values that satisfy all equations simultaneously. Consider the given exercise where each part is a different system of two equations in two variables, namely \(x_1\) and \(x_2\).

There are various methods to solve these systems, such as substitution, elimination, and graphing. Cramer's Rule, used in the exercise, is a direct method utilizing determinants, making it especially useful for systems that can be represented with square matrices. Each variable solution is expressed as a fraction where the numerator is a determinant of a matrix modified by replacing one column with the constant terms, and the denominator is the determinant of the coefficient matrix.

Solving such systems using Cramer's Rule is advantageous for handling it systematically once you accurately compute the determinants. It's important, though, to ensure the determinant of the coefficient matrix (main determinant) is non-zero; otherwise, the system may be inconsistent or have infinitely many solutions.

Determinants

Determinants are special numbers you can calculate from a square matrix. They provide important properties of the matrix and are crucial in solving linear equations, particularly through methods like Cramer's Rule.

To explain, consider a 2x2 matrix represented as follows:\[A = \begin{bmatrix}a_{11} & a_{12} \a_{21} & a_{22}\end{bmatrix}\] The determinant of matrix \(A\), denoted as \(|A|\), is calculated using this formula: \[|A| = a_{11}a_{22} - a_{12}a_{21}\].

Determinants help determine the nature of the system solutions:

• If the determinant is zero, the system might not have a unique solution. It could be either dependent (leading to infinitely many solutions) or inconsistent (no solutions).
• If the determinant is non-zero, it implies a unique solution exists.

Understanding how to compute and interpret determinants is essential in linear algebra, especially when applying Cramer's Rule.

Linear Algebra

Linear Algebra is a branch of mathematics centered around vector spaces and linear transformations between vector spaces. It is the mathematical framework we use to handle systems of linear equations and matrix operations.

In solving the exercise, linear algebra provides the structure through which we interpret and solve systems of equations efficiently. By representing systems with matrices, we leverage the power of matrix theory to find solutions using determinant-based methods, such as Cramer's Rule.

The methods and concepts in linear algebra are not just theoretical; they are widely applied in numerous fields such as physics, computer science, engineering, economics, and more. Understanding linear algebra helps simplify complex systems into manageable mathematical computations.

Matrix Theory

Matrix Theory is a pillar of linear algebra, focusing extensively on matrices, their properties, and operations. In the context of solving systems of equations, matrices allow for a compact, efficient representation of linear systems.

A matrix is essentially a rectangular array of numbers, organized in rows and columns. The exercise used matrices to encapsulate the coefficients of variables and the constants as well. With Cramer's Rule, the coefficient matrix and the matrices formed by replacing columns with constant terms were central to finding the solutions, leveraging the power of determinants.

Here's why understanding matrices is crucial:

• Representation: Matrices provide a consistent and visually intuitive way of representing complex systems.
• Operations: Fundamental operations like addition, multiplication, and finding determinants are built on matrix rules.
• Application: Many applied mathematics and engineering problems naturally form matrices for analysis and solutions.

Comprehending matrix theory is foundational for anyone delving deeper into linear algebra and related fields.