Problem 44 - College Algebra

Question

Write the system of linear equations for which Cramer's Rule yields the given determinants. $$D=\left|\begin{array}{rr}2 & -3 \\\5 & 6\end{array}\right|, \quad D_{x}=\left|\begin{array}{rr}8 & -3 \\\11 & 6\end{array}\right|$$

Step-by-Step Solution

Answer

The system of linear equations for which Cramer's Rule yields the given determinants is \(2x - 3y = 8\) and \(5x + 6y = 11\).

Step 1: Understand Cramer's Rule

Cramer's Rule states that in a system of linear equations, the value of the unknowns (let's call them x and y) can be found by \(x = D_x / D\) and \(y = D_y / D\). Here, D is the determinant of the coefficient matrix, \(D_x\) is the determinant of the matrix formed by replacing the x-coefficients with the constants, and similarly for \(D_y\). The coefficients of x and y in the system of equations can be found in the determinant D.

Step 2: Identify the coefficients of the system from the matrix D

The determinant D, \(|2,-3;5,6|\), represents the coefficient matrix of the system. Therefore, the system of linear equations can be written as \(2x - 3y = a\) and \(5x + 6y = b\), where a and b are the constants we need to find.

Step 3: Identify the constants of the system from the matrix \(D_x\)

The determinant \(D_x\), \(|8,-3;11,6|\), represents the matrix formed by replacing the x-coefficients with the constants. Therefore, the system of linear equations now can be written as \(8x - 3y = a\) and \(11x + 6y = b\). Comparing these equations to the ones found in Step 2, we can find that \(a = 8\) and \(b = 11\). The system of equations becomes \(2x - 3y = 8\) and \(5x + 6y = 11\).

Key Concepts

Determinants

Determinants are special numbers that can be calculated from a square matrix. They play a crucial role in mathematics, especially in solving systems of linear equations and in the field of matrix algebra.
Determinants serve multiple purposes, such as determining if a matrix is invertible or calculating a matrix's eigenvalues. However, in the context of Cramer's Rule, determinants help solve linear equations.
For a 2x2 matrix, the determinant is computed as follows: if you have a matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is \( ad - bc \).
In Cramer's Rule, the determinant of the coefficient matrix \( D \) is compared to the determinants \( D_x \) and \( D_y \), which are derived by replacing one of the columns of \( D \) with the constants from the equations. These replacements help isolate and solve for specific variables in the equations.

System of Linear Equations

A system of linear equations is a set of two or more linear equations with the same variables. The goal is to find the values of these variables that satisfy all the equations simultaneously.
For example, consider a system with two equations as follows:

• \( 2x - 3y = 8 \)
• \( 5x + 6y = 11 \)

To solve this system using Cramer's Rule, it involves representing the equations in matrix form. The coefficients of the variables form the coefficient matrix, and a solution can be found by applying the determinants.
Each equation in the system represents a line on a graph. The solution to the system is the point at which these lines intersect. If the lines are parallel, there is no solution; if the same line is represented, there are infinitely many solutions.

Matrix Algebra

Matrix algebra involves operations like addition, subtraction, and multiplication of matrices. It is a branch of mathematics dealing with matrices and their properties.
Matrices are rectangular arrays of numbers, and they can represent systems of linear equations. Matrix algebra simplifies complex sets of equations into manageable operations. This is especially useful in solving linear equations using tools like Cramer's Rule.
In our previous example, the coefficient matrix \( \begin{pmatrix} 2 & -3 \ 5 & 6 \end{pmatrix} \) represents the given system's coefficients. Using matrix algebra, you can perform operations to find determinants, inverses, and other essential information that helps solve systems of equations.
The ability to manipulate matrices and understand their properties is fundamental in many mathematical applications, from computer graphics to theoretical physics.