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Chapter 9: Problem 56

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 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 by step solution

01

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.
02

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 \(D_{x} = det \left(\begin{array}{cc} m & b \\ n & d \end{array}\right) = md - bn\).
03

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 \(D_{y} = det \left(\begin{array}{cc} a & m \\ c & n \end{array}\right) = an - cm\).

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Most popular questions from this chapter

Perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} {4} & {0} \\ {-3} & {5} \\ {0} & {1} \end{array}\right] \quad B=\left[\begin{array}{rr} {5} & {1} \\ {-2} & {-2} \end{array}\right] \quad C=\left[\begin{array}{rr} {1} & {-1} \\ {-1} & {1} \end{array}\right] $$ $$ A(C B) $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I added matrices of the same order by adding corresponding elements.

Explaining the Concepts Describe how to use row operations and matrices to solve a system of linear equations.

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{aligned} 3 w-4 x+y+z &=9 \\ w+x-y-z &=0 \\ 2 w+x+4 y-2 z &=3 \\ -w+2 x+y-3 z &=3 \end{aligned}\right. $$

Write a system of linear equations in three or four variables to solve. Then use matrices to solve the system. Imagine the entire global population as a village of precisely 200 people. The bar graph shows some numeric observations based on this scenario. (graph can't copy) Combined, there are 183 Asians, Africans, Europeans, and Americans in the village. The number of Asians exceeds the number of Africans and Europeans by \(70 .\) The difference between the number of Europeans and Americans is \(15 .\) If the number of Africans is doubled, their population exceeds the number of Europeans and Americans by \(23 .\) Determine the number of Asians, Africans, Europeans, and Americans in the global village.
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