Problem 52 - College Algebra

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 determinants \(D_{x}\) and \(D_{y}\) are calculated using the coefficients and constants of the system. \(D_{x}\) is given by \(D_{x} = ed - bf\) and \(D_{y}\) is given by \(D_{y} = af - ce\), where \(a, b, c, d\) are the coefficients and \(e, f\) are the constants of the system.

Step 1: Write the system in matrix form

Any system of equations can be written in the form of a matrix. Consider the system of equations as \(ax + by = e\) & \(cx + dy = f\). This can be written as a matrix as follows: \[ \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \left[ \begin{array}{c} x \ y \end{array} \right] = \left[ \begin{array}{c} e \ f \end{array} \right] \] At this point, we should familiarize ourselves with the calculation of the determinant of a 2x2 matrix. The determinant \(D\) of matrix \[ A = \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \] is \( D = ad - bc \). This will be useful for the next steps.

Step 2: Calculate the main determinant - D

Determine the main determinant \(D\) of the system replacing \(x, y\) with coefficients \(a, b, c, d\) as explained in step 1. Thus, the determinant \(D\) of this system is: \(D = ad - bc\).

Step 3: Calculate the determinant Dx

The determinant \(D_{x}\) is obtained by replacing the first column of \(D\) (coefficient of x) with the constants from the right side of the system (the constants \(e, f\)). Therefore, determinant \(D_{x}\) of this system is calculated as: \(D_{x} = ed - bf)\.

Step 4: Calculate the determinant Dy

In the same way, \(D_{y}\) is obtained by replacing the second column of \(D\) (the coefficient of y) with the constants from the right side of the system (the constants \(e, f\)). So, determinant \(D_{y}\) of this system is: \(D_{y} = af - ce\).