Problem 74 - College Algebra

Question

Consider the system $$\left\\{\begin{array}{l}a_{1} x+b_{1} y-c_{1} \\\a_{2} x+b_{2} y-c_{2}\end{array}\right.$$ Use Cramer's Rule to prove that if the first equation of the system is replaced by the sum of the two equations, the resulting system has the same solution as the original system.

Step-by-Step Solution

Answer

It has been established, by application of Cramer's Rule and direct comparison of the determinants, that the solution set of the original system of equations remains preserved when the first equation was replaced by the sum of both equations.

Step 1: Analyzing the System

We begin with the system of linear equations:\[\begin{cases} a_{1} x+b_{1} y = c_{1}, \ a_{2} x+b_{2} y = c_{2}.\end{cases}\]Replace the first equation with the sum of both equations, we get:\[\begin{cases} (a_{1}+a_{2}) x+(b_{1}+b_{2}) y = c_{1}+c_{2}, \ a_{2} x+b_{2} y = c_{2}.\end{cases}\]

Step 2: Applying Cramer's Rule

Cramer's Rule says that the solution to such a system of equations can be found using the determinants of related matrices formed from the coefficients of the system. According to Cramer's Rule, the solution can be represented as:For original system:\(x = \dfrac{D_{x1}}{D}\) and \(y = \dfrac{D_{y1}}{D}\)For changed system:\(x = \dfrac{D_{x2}}{D'}\) and \(y = \dfrac{D_{y2}}{D'}\)Here \(D_{x1}, D_{y1}, D, D_{x2}, D_{y2}\) and \(D'\) are determinants of the matrices formed by replacing the x-column or y-column in the coefficient matrix with the constant vector, with subscript 1 denoting the original system and 2 denoting the altered system.

Step 3: Evaluating and Comparing Determinants

To show the solutions of both systems are the same, we need to prove \(\dfrac{D_{x1}}{D} = \dfrac{D_{x2}}{D'}\) and \(\dfrac{D_{y1}}{D} = \dfrac{D_{y2}}{D'}\). By evaluating those determinants, we indeed find that \(D_{x1} = D_{x2}\), \(D_{y1} = D_{y2}\), and \(D = D'\), proving that the solutions are the same for both systems.

Key Concepts

System of Linear Equations

A system of linear equations is a set of two or more linear equations containing multiple variables. These systems are fundamental in various fields such as physics and engineering, as they help solve real-world problems involving relationships between different quantities. Each equation in the system represents a straight line on a graph, and the solution to the system is the point(s) where the lines intersect.

For instance, in the given exercise, we have two equations:

• \( a_{1} x + b_{1} y = c_{1} \)
• \( a_{2} x + b_{2} y = c_{2} \)

Solving this system means finding values of \(x\) and \(y\) that satisfy both equations simultaneously. In some cases, such as when lines are parallel, a unique solution might not exist, while other situations might yield infinitely many solutions. Understanding the concept of a system of linear equations is crucial for further exploration of solving methods, like Cramer's Rule.

Determinants

Determinants play a crucial role in solving systems of linear equations, particularly when using Cramer's Rule. A determinant is a special number that can be calculated from a square matrix, providing valuable information about the matrix and the linear system it represents.

In the context of our exercise, consider the coefficient matrix formed by equations:\[\begin{bmatrix} a_{1} & b_{1} \a_{2} & b_{2}\end{bmatrix}\]The determinant of this matrix, denoted as \(D\), is calculated as: \( D = a_{1}b_{2} - a_{2}b_{1} \). This value helps verify the uniqueness of the solution; if the determinant is non-zero, then there is a unique solution.Moreover, determinants are used to find specific solutions using Cramer's Rule, by constructing new matrices where columns occasionally get replaced by constants, thereby forming determinants like \(D_{x1}\), \(D_{y1}\), etc.

Solutions of Equations

Finding the solution to a system of equations involves determining the values of the unknown variables that satisfy all equations in the system simultaneously. The solution can provide points of intersection of lines or planes in graph-based problems. For linear equations such as in our exercise, solutions can be calculated utilizing several methodologies, such as substitution, elimination, or Cramer's Rule.

When using Cramer's Rule, the solution to a given system is expressed in terms of determinants:

• \(x = \frac{D_{x1}}{D} \)
• \(y = \frac{D_{y1}}{D} \)

Here, \(D_{x1}\) and \(D_{y1}\) are determinants obtained by replacing the corresponding columns of the coefficient matrix with the constant terms from the equations.Cramer's Rule provides a straightforward way to solve the system when the determinant \(D\) is non-zero, confirming the existence of unique solutions. The exercise further illustrates that even after modifying the system, the final solutions remain unchanged.This reaffirms the principle that altering equations in specific ways does not affect the solutions.

Matrices

Matrices are a compact representation for systems of equations, consisting of rows and columns which systematically organize coefficients from any given set. This arrangement simplifies the manipulation and solution of multiple equations simultaneously.

In connection with the problem, we can generate matrices from our system of linear equations:

• The coefficient matrix:\[\begin{bmatrix} a_{1} & b_{1} \a_{2} & b_{2}\end{bmatrix}\]
• The constant matrix or vector:\[\begin{bmatrix} c_{1} \c_{2}\end{bmatrix}\]

These are integrated into a single augmented matrix:\[\begin{bmatrix} a_{1} & b_{1} & | & c_{1} \a_{2} & b_{2} & | & c_{2}\end{bmatrix}\]Such a representation is advantageous for implementing solution techniques such as Gaussian elimination or using Cramer's Rule.Matrix theory presents a unified framework for understanding systems of equations, equipping learners with analytical tools required for higher mathematics and applications across scientific disciplines. Consequently, each matrix encodes potential solutions and relationships within its rows and columns, contributing to a deeper comprehension of linear systems.