Problem 56 - Algebra and Trigonometry

Question

Describe the determinants \(D_{x}\) and \(D_{y}\) in terms of the coefficients and constants in a system of two equations in two variables.

Step-by-Step Solution

Answer

The determinant \(D_{x}\) is calculated as \(md - bn\), by replacing the coefficients of x with the constants in the determinant of the system. The determinant \(D_{y}\) is calculated as \(an - cm\), by replacing the coefficients of y with the constants in the determinant of the system.

Step 1: Understanding the System

A typical simultaneous linear system in two variables can be represented as \(ax + by = m\) (equation 1) and \(cx + dy = n\) (equation 2). The coefficients of x and y (i.e., a, b, c, and d) and constants m and n are the primary constituents of the determinant.

Step 2: Calculating \(D_{x}\)

The determinant \(D_{x}\) is obtained by replacing the coefficients of x in the original determinant with the constants (m and n). The determinant is calculated as \(D_{x} = det \left(\begin{array}{cc} m & b \\ n & d \end{array}\right) = md - bn\).

Step 3: Calculating \(D_{y}\)

The determinant \(D_{y}\) is obtained by replacing the coefficients of y in the original determinant with the constants (m and n). The determinant is calculated as \(D_{y} = det \left(\begin{array}{cc} a & m \\ c & n \end{array}\right) = an - cm\).