Problem 103 - Algebra and Trigonometry

Question

Use Cramer's Rule (determinants) to solve the system: $$ \left\\{\begin{aligned} x-y &=-5 \\ 3 x+2 y &=0 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solution to the system is x = -2 and y = 3

Step 1: Calculating Determinant D

Determine matrix D by writing the coefficients of the variables in the given system of equations:\[ D = \begin{bmatrix} 1 & -1 \ 3 & 2 \end{bmatrix} \]. The determinant (D) is calculated by subtracting the product of the diagonals: D = (1)(2) - (-1)(3) = 2+3 = 5

Step 2: Calculating Determinant \(D_x\)

To find determinant \(D_x\), replace the first column in the matrix D with the constants from the right side of the equations and calculate the determinant in the same way as Step 1:\[ D_x = \begin{bmatrix} -5 & -1 \ 0 & 2 \end{bmatrix} \]. The determinant \(D_x\) = (-5)(2) - (-1)(0) = -10

Step 3: Calculating Determinant \(D_y\)

To find determinant \(D_y\), replace the second column in the matrix D with the constants from the right side of the equations and calculate the determinant in the same way as Step 1:\[ D_y = \begin{bmatrix} 1 & -5 \ 3 & 0 \end{bmatrix} \]. The determinant \(D_y\) = (1)(0) - (-5)(3) = 15

Step 4: Finding x and y using Cramer's Rule

The solutions according to Cramer's Rule can be found by dividing determinants \(D_x\) and \(D_y\) by D. Thus, \(x = D_x/D = -10/5 = -2\) and \(y = D_y/D = 15/5 = 3\)

Key Concepts

Understanding Determinants

Determinants are a key concept in linear algebra. They are special numbers that can be calculated from a square matrix. You can think of a determinant as a function that provides crucial information about the matrix and its properties.
For a 2x2 matrix, the determinant is found by subtracting the product of the diagonals:

• If the matrix is \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), then the determinant is \( ad - bc \).

Determinants are important because they help to determine if a system of equations has a unique solution. If the determinant is zero, the system might not have a unique solution. In the context of solving equations, the determinant tells us whether the matrix has an inverse and, thus, whether Cramer's Rule can be used.

Exploring Systems of Equations

A system of equations consists of multiple equations that are solved together. Each equation shares the same set of variables.
Consider the system given in the exercise:

• \( x - y = -5 \)
• \( 3x + 2y = 0 \)

To solve these, we need to find values for \( x \) and \( y \) that satisfy both equations simultaneously. Methods to solve these include substitution, elimination, and using matrix algebra like Cramer's Rule. The goal is to find a common solution that works for all the equations in the system.

Basics of Matrix Algebra

Matrix algebra involves operations with matrices, which are rectangular arrays of numbers.
Matrices can be added, subtracted, and multiplied, and these operations can be used to solve systems of equations. When solving with matrices, the coefficients of variables in a system of equations form a matrix. For instance, the coefficients in the system \( (x-y=-5, 3x+2y=0) \) form the matrix:

• \( \begin{bmatrix} 1 & -1 \ 3 & 2 \end{bmatrix} \)

Matrix algebra simplifies solving larger systems and allows the use of operations that are systematic and computationally efficient. Cramer's Rule is a technique within matrix algebra that utilizes determinants to find solutions for variables.

Solving Linear Equations with Cramer's Rule

Cramer's Rule is a straightforward method used to solve systems of linear equations using determinants.
This rule is applicable only when the number of equations is the same as the number of unknowns, and the determinant of the coefficients matrix is non-zero. Here's how it works:

• First, calculate the determinant \( D \) of the coefficient matrix.
• Next, find \( D_x \) by replacing the first column of the original matrix with the constants from the equations.
• Determine \( D_y \) by replacing the second column with those constants.
• Finally, compute the solution as \( x = D_x/D \) and \( y = D_y/D \).

In our example, these steps led us to find \( x = -2 \) and \( y = 3 \). This method is effective for systems that are two or three equations in size, providing a clear geometric interpretation of the solution.