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Chapter 6: Problem 52

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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: \(D = ad - bc\).
03

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)\.
04

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: \(D_{y} = af - ce\).

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Most popular questions from this chapter

In Exercises \(1-4\) a. Give the order of each matrix. b. If \(A=\left[a_{i j}\right],\) identify \(a_{32}\) and \(a_{23}\) or explain why identification is not possible. $$ \left[\begin{array}{rrrr} -4 & 1 & 3 & -5 \\ 2 & -1 & \pi & 0 \\ 1 & 0 & -e & \frac{1}{5} \end{array}\right] $$

Determinants are used to find the area of a triangle whose vertices are given by three points in a rectangular coordinate system. The area of a triangle with vertices\(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) is $$ \text { Area }=\pm \frac{1}{2}\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1 \\\x_{3} & y_{3} & 1\end{array}\right| $$ where the \(\pm\) symbol indicates that the appropriate sign should be chosen to yield a positive area. Use this information to work. Use determinants to find the area of the triangle whose vertices are \((1,1),(-2,-3),\) and \((11,-3)\).

Find two matrices \(A\) and \(B\) such that \(A B=B A\)

We have seen that determinants can be used to solve linear equations, give areas of triangles in rectangular coordinates, and determine equations of lines. Not impressed with these applications? Members of the group should research an application of determinants that they find intriguing. The group should then present a seminar to the class about this application.

In Exercises \(27-36,\) find (if possible): \(\begin{array}{llll}\text { a. } A B & \text { and } & \text { b. } B A\end{array}\) $$ A=\left[\begin{array}{l} -1 \\ -2 \\ -3 \end{array}\right], \quad B=\left[\begin{array}{lll} 1 & 2 & 3 \end{array}\right] $$
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