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Chapter 2: Problem 37

Write the given system of linear equations in matrix form. $$ \begin{array}{r} 2 x-3 y+4 z=6 \\ 2 y-3 z=7 \\ x-y+2 z=4 \end{array} $$

Short Answer

Expert verified
The given system of linear equations can be written in matrix form as: $$ \begin{bmatrix} 2 & -3 & 4 \\ 0 & 2 & -3 \\ 1 & -1 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 7 \\ 4 \end{bmatrix} $$

Step by step solution

01

Identify the coefficients and constants

The given system of linear equations is: $$ \begin{array}{r} 2x - 3y + 4z = 6 \\ 2y - 3z = 7 \\ x - y + 2z = 4 \end{array} $$ The coefficients of x, y, and z are 2, -3, and 4 in the first equation, 0, 2, and -3 in the second equation, and 1, -1, and 2 in the third equation, respectively. The constants on the right-hand side are 6, 7, and 4.
02

Write down the matrix of coefficients

Now we can write the matrix A, which includes the coefficients of the variables x, y, and z from the given system of linear equations: $$ A = \begin{bmatrix} 2 & -3 & 4 \\ 0 & 2 & -3 \\ 1 & -1 & 2 \end{bmatrix} $$
03

Provide the matrix of constants

Next, let's write the matrix b, which includes the constants 6, 7, and 4: $$ b = \begin{bmatrix} 6 \\ 7 \\ 4 \end{bmatrix} $$
04

Combine the matrices into Matrix form

Lastly, we can combine the matrices A and b, along with the column vector x consisting of the variables x, y, and z, to put the system of linear equations into matrix form Ax = b: $$ \begin{bmatrix} 2 & -3 & 4 \\ 0 & 2 & -3 \\ 1 & -1 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 7 \\ 4 \end{bmatrix} $$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
Linear equations are equations where each term is either a constant or the product of a constant with a single variable. They're fundamental in mathematics as they represent the simplest form of relationships between different variables. In our given system, we have three equations each involving variables \(x\), \(y\), and \(z\). Each equation represents a straight line in a three-dimensional space.Understanding linear equations:
  • Each equation in the system represents a plane in a 3D space when we have three variables.
  • The solutions are the points where these planes intersect.
  • In the simplest form, a linear equation in two variables is often expressed as \(ax + by = c\).
In more complex systems, linear equations can be extended to any number of variables, forming a system of linear equations. The primary goal in solving these systems is to find values for the variables that satisfy all given equations simultaneously.
Matrix Operations
Matrix operations are key tools in structuring and solving systems of linear equations. They help organize data in a manageable way and provide techniques to solve equations efficiently.Using matrices:
  • Matrices are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns.
  • In our exercise, the matrix \(A\) represents the coefficients of the system of equations, capturing all equations' key components in one single entity.
  • Operations like matrix addition, subtraction, and multiplication can help in solving and simplifying systems of equations.
Particularly useful in larger systems, matrix operations such as row elimination or determinant calculation can make finding solutions faster and more straightforward. In the matrix form \(Ax = b\), 'A' consists of coefficients, 'x' is the column vector of variables, and 'b' is the column vector with constants from the equations. This arrangement simplifies complex computations.
System of Equations
A system of equations involves finding multiple unknowns through a set of equations. Such systems often arise in real-life scenarios where multiple conditions simultaneously constrain results. Solving systems:
  • The system is considered "consistent" if a solution exists and "inconsistent" if no solution exists.
  • "Dependent" systems have infinitely many solutions, while "independent" systems have a unique solution.
  • The given system can be effectively solved using matrix forms and operations, which allows for algebraic methods such as Gaussian elimination or matrix inversion.
By placing the system into matrix form, the process integrates algebraic approaches with computational efficiency. The matrix representation not only organizes the equations but also enables the application of powerful computational techniques to find solutions more systematically.

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Most popular questions from this chapter

The problems in exercise correspond to those in exercises 15-27, Section 2.1. Use the results of your previous work to help you solve these problems. The annual returns on Sid Carrington's three investments amounted to $$\$ 21,600$$: \(6 \%\) on a savings account, \(8 \%\) on mutual funds, and \(12 \%\) on bonds. The amount of Sid's investment in bonds was twice the amount of his investment in the savings account, and the interest earned from his investment in bonds was equal to the dividends he received from his investment in mutual funds. Find how much money he placed in each type of investment.

Bob, a nutritionist who works for the University Medical Center, has been asked to prepare special diets for two patients, Susan and Tom. Bob has decided that Susan's meals should contain at least \(400 \mathrm{mg}\) of calcium, \(20 \mathrm{mg}\) of iron, and \(50 \mathrm{mg}\) of vitamin \(\mathrm{C}\). whereas Tom's meals should contain at least \(350 \mathrm{mg}\) of calcium, \(15 \mathrm{mg}\) of iron, and \(40 \mathrm{mg}\) of vitamin \(\mathrm{C}\). Bob has also decided that the meals are to be prepared from three basic foods: food \(\mathrm{A}\), food \(\mathrm{B}\), and food \(\mathrm{C}\). The special nutritional contents of these foods are summarized in the accompanying table. Find how many ounces of each type of food should be used in a meal so that the minimum requirements of calcium, iron, and vitamin \(\mathrm{C}\) are met for each patient's meals. $$ \begin{array}{lccc} \hline && {\text { Contents (mg/oz) }} & \\ & \text { Calcium } & \text { Iron } & \text { Vitamin C } \\ \hline \text { Food A } & 30 & 1 & 2 \\ \hline \text { Food B } & 25 & 1 & 5 \\ \hline \text { Food C } & 20 & 2 & 4 \\ \hline \end{array} $$

Kaitlin and her friend Emma returned to the United States from a tour of four cities: Oslo, Stockholm, Copenhagen, and Saint Petersburg. They now wish to exchange the various foreign currencies that they have accumulated for U.S. dollars. Kaitlin has 82 Norwegian krones, 68 Swedish krones, 62 Danish krones, and 1200 Russian rubles. Emma has 64 Norwegian krones, 74 Swedish krones, 44 Danish krones, and 1600 Russian rubles. Suppose the exchange rates are U.S. \(\$ 0.1651\) for one Norwegian krone, U.S. \$0.1462 for one Swedish krone, U.S. \$0.1811 for one Danish krone, and U.S. \(\$ 0.0387\) for one Russian ruble. a. Write a \(2 \times 4\) matrix \(A\) giving the values of the various foreign currencies held by Kaitlin and Emma. (Note: The answer is not unique.) b. Write a column matrix \(B\) giving the exchange rate for the various currencies. c. If both Kaitlin and Emma exchange all their foreign currencies for U.S. dollars, how many dollars will each have?

Solve the system of linear equations using the Gauss-Jordan elimination method. $$ \begin{array}{rr} 2 x+3 z= & -1 \\ 3 x-2 y+z= & 9 \\ x+y+4 z= & 4 \end{array} $$

(a) write each system of equations as a matrix equation and (b) solve the system of equations by using the inverse of the coefficient matrix. $$ \begin{array}{l} \begin{aligned} x_{1}+x_{2}+2 x_{3}+x_{4} &=b_{1} \\ 4 x_{1}+5 x_{2}+9 x_{3}+x_{4} &=b_{2} \\ 3 x_{1}+4 x_{2}+7 x_{3}+x_{4} &=b_{3} \\ 2 x_{1}+3 x_{2}+4 x_{3}+2 x_{4} &=b_{4} \end{aligned}\\\ \text { where } \quad \text { (i) } b_{1}=3, b_{2}=6, b_{3}=5, b_{4}=7\\\ \text { and (ii) } b_{1}=1, b_{2}=-1, b_{3}=0, b_{4}=-4 \end{array} $$
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