Problem 24 - Algebra 2

Question

Use Cramer’s Rule to solve each system of equations. \(0.5 r-s=-1\) \(0.75 r+0.5 s=-0.25\)

Step-by-Step Solution

Answer

The solution is \( r = -0.75 \) and \( s = 0.625 \).

Step 1: Write the system in matrix form

The given system of equations is:1. \(0.5r - s = -1\)2. \(0.75r + 0.5s = -0.25\)In matrix form, this can be written as:\[\begin{bmatrix} 0.5 & -1 \ 0.75 & 0.5 \end{bmatrix} \begin{bmatrix} r \ s \end{bmatrix} = \begin{bmatrix} -1 \ -0.25 \end{bmatrix}\]

Step 2: Find the determinant of the coefficient matrix

The coefficient matrix is:\[\begin{bmatrix} 0.5 & -1 \ 0.75 & 0.5 \end{bmatrix}\]To find the determinant, use the formula for a 2x2 matrix:\[\det(A) = (0.5)(0.5) - (-1)(0.75) = 0.25 + 0.75 = 1\]

Step 3: Create matrix for r and find its determinant

Replace the first column of the coefficient matrix with the constants from the right-hand side:\[\begin{bmatrix} -1 & -1 \ -0.25 & 0.5 \end{bmatrix}\]Now, find its determinant:\[\det(A_r) = (-1)(0.5) - (-1)(-0.25) = -0.5 + 0.25 = -0.75\]

Step 4: Create matrix for s and find its determinant

Replace the second column of the coefficient matrix with the constants from the right-hand side:\[\begin{bmatrix} 0.5 & -1 \ 0.75 & -0.25 \end{bmatrix}\]Now, find its determinant:\[\det(A_s) = (0.5)(-0.25) - (-1)(0.75) = -0.125 + 0.75 = 0.625\]

Step 5: Solve for r using Cramer's Rule

According to Cramer's Rule, the solution for \(r\) is given by:\[r = \frac{\det(A_r)}{\det(A)} = \frac{-0.75}{1} = -0.75\]

Step 6: Solve for s using Cramer's Rule

According to Cramer's Rule, the solution for \(s\) is given by:\[s = \frac{\det(A_s)}{\det(A)} = \frac{0.625}{1} = 0.625\]

Key Concepts

Systems of Equations

In mathematics, a system of equations is a collection of two or more equations with the same set of unknowns. In our exercise, there are two equations that are interconnected:

• 0.5r - s = -1
• 0.75r + 0.5s = -0.25

Each equation consists of variables, here represented by \(r\) and \(s\), and constants. The objective is to find the values of these variables that simultaneously satisfy both equations. Such systems are often encountered when dealing with real-world problems where multiple relationships exist simultaneously.
The solution to a system of equations can be approached through various methods. One such method is Cramer's Rule, which applies when the system can be represented by a square matrix and when the determinant is non-zero.

2x2 Matrix

The term 2x2 matrix refers to a grid containing four elements arranged in two rows and two columns. When dealing with systems of linear equations, matrices serve as a compact notation. Let's look at our system again:

• The coefficients from the equations arrange into a 2x2 matrix:
• \[ \begin{bmatrix} 0.5 & -1 \ 0.75 & 0.5 \end{bmatrix} \]

In this matrix, the first column represents the coefficients of variable \(r\), and the second column represents the coefficients of \(s\).
Matrices are useful for performing operations such as addition, subtraction, and especially multiplication, which is essential for solving more complex systems. They provide an organized framework when working through the step-by-step processes involved in solving systems of equations.

Determinant

The determinant is a special number calculated from a square matrix. It plays a crucial role in determining whether a system of linear equations has a unique solution. In our case, we calculate the determinant of the coefficient matrix \(A\):

• \( \det(A) = 0.5 \times 0.5 - (-1) \times 0.75 = 0.25 + 0.75 = 1 \)

This calculation involves multiplying diagonally across the matrix, following the formula for a 2x2 matrix determinant. The determinant being non-zero (here, \(1\)) indicates that the system of equations has a unique solution.
Determinants also help in using Cramer's Rule where the determinant of matrices with modified columns are used to find the solution variables \(r\) and \(s\).

Linear Algebra

Linear Algebra is a branch of mathematics that focuses extensively on solving systems of linear equations, vector spaces, and matrix transformations. It's all about finding systematic ways to solve mathematical problems involving lines, planes, and transformations.
In practical terms, linear algebra is used in fields such as physics, engineering, computer science, and economics, wherever complex systems and multidimensional data sets are explored. The exercise provided is a classic linear algebra problem, illustrating how matrices and determinants work together to solve systems of equations using Cramer's Rule, highlighting the synergy between abstract concepts and their real-world applications.