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Chapter 2: Problem 5

Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations, (a) determine whether the system has a solution and (b) find the solution or solutions to the system, if they exist. $$ \left[\begin{array}{rrr|r} 1 & 0 & 1 & 4 \\ 0 & 1 & 0 & -2 \end{array}\right] $$

Short Answer

Expert verified
The given row-reduced augmented matrix represents a consistent system of linear equations. The solution to the system is: \(\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 4-t \\ -2 \\ t \end{bmatrix}\), where \(t\) is a free parameter. Since there is a free parameter in the solution, the system has infinitely many solutions.

Step by step solution

01

Analyze the augmented matrix

We are given the following row-reduced augmented matrix: \[ \left[\begin{array}{rrr|r} 1 & 0 & 1 & 4 \\\ 0 & 1 & 0 & -2 \end{array}\right] \] The matrix represents the following system of linear equations: 1. \(x_1 + x_3 = 4\) 2. \(x_2 = -2\)
02

Determine if the system has a solution

There are no inconsistencies in the augmented matrix, and all rows in the matrix correspond to valid equations (no rows are of the form 0 = non-zero constant). This means that the system is consistent, and it has a solution.
03

Find the solution(s)

Since our system is consistent, we now need to find the solution(s) to the system. Observing the equations, we can see that the variables \(x_1\) and \(x_2\) can be expressed in terms of \(x_3\). Therefore, we can write the solution as a parametric expression. Let \(x_3 = t\), where t is a parameter. Then we have: - From equation 1: \(x_1 = 4 - t\). - From equation 2: \(x_2 = -2\). The solution to the given system is: \(\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 4-t \\ -2 \\ t \end{bmatrix}\), where \(t\) is a free parameter. Since there is a free parameter in the solution, the system has infinitely many solutions.

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Key Concepts

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

System of Linear Equations
A system of linear equations is a collection of one or more linear equations involving the same set of variables. For example, a simple system might consist of two equations and two unknowns. These systems can be represented in several ways, including graphically on a coordinate plane, algebraically with equations, or more compactly with matrices鈥攁 method especially useful for systems with more than two equations.

A row-reduced augmented matrix is a matrix that has been simplified using elementary row operations, making it easier to determine solutions for the associated system of linear equations. The left portion before the vertical bar represents the coefficients of the variables in each equation, while the right portion contains the constants from the right side of the equations. The given row-reduced augmented matrix:
\[\left[\begin{array}{rrr|r}1 & 0 & 1 & 4 \ 0 & 1 & 0 & -2 \end{array}\right]\]
encapsulates a system of two linear equations with three variables (\(x_1, x_2, x_3\)).
Parametric Expression
In the context of linear algebra, a parametric expression is used to describe the solutions of a system of linear equations when it has at least one free variable鈥攁 variable that isn't led by a leading 1 in any row of a row-reduced matrix. This expression allows us to show all possible solutions using parameters, which are variables that take on an infinite number of values.

When dealing with a row-reduced augmented matrix, identifying a free variable is crucial, as it signifies the presence of infinitely many solutions. In the step-by-step solution given for our system, the variable \(x_3\) is the free variable, chosen to be represented by the letter 't'. By expressing \(x_1\) and \(x_2\) in terms of 't', we encapsulate a wealth of possibilities, with each value of 't' giving us a different solution to the system.
Infinitely Many Solutions
When a system of linear equations is said to have infinitely many solutions, it means exactly that鈥攖here's an endless set of different solutions or points that satisfy all of the equations in the system. This typically occurs when the system is underdetermined, meaning there are more variables than independent equations to constrain them, or when multiple equations describe the same line or plane.

The presence of free variables, as observed in the row-reduced augmented matrix from our exercise, directly signals that the system has infinitely many solutions. Each value of the free variable 't' generates a unique solution. Thus, instead of a single point of intersection, we have a line or even a plane of intersecting points, showcasing the limitless myriad of solutions.

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Most popular questions from this chapter

Mr. and Mrs. Garcia have a total of \(\$ 100,000\) to be invested in stocks, bonds, and a money market account. The stocks have a rate of return of \(12 \% /\) year, while the bonds and the money market account pay \(8 \% /\) year and \(4 \% /\) year, respectively. The Garcias have stipulated that the amount invested in stocks should be equal to the sum of the amount invested in bonds and 3 times the amount invested in the money market account. How should the Garcias allocate their resources if they require an annual income of \(\$ 10,000\) from their investments? Give two specific options.

A university admissions committee anticipates an enrollment of 8000 students in its freshman class next year. To satisfy admission quotas, incoming students have been categorized according to their sex and place of residence. The number of students in each category is given by the matrix $$ \begin{array}{l} \text { In-state } \\ \text { A= Out-of-state } \\ \text { Foreign } \end{array}\left[\begin{array}{rr} 2700 & 3000 \\ 800 & 700 \\ 500 & 300 \end{array}\right] $$ By using data accumulated in previous years, the admissions committee has determined that these students will elect to enter the College of Letters and Science, the College of Fine Arts, the School of Business Administration, and the School of Engineering according to the percentages that appear in the following matrix: $$ B=\begin{array}{l} \text { Male } \\ \text { Female } \end{array}\left[\begin{array}{llll} 0.25 & 0.20 & 0.30 & 0.25 \\ 0.30 & 0.35 & 0.25 & 0.10 \end{array}\right] $$ Find the matrix \(A B\) that shows the number of in-state, outof-state, and foreign students expected to enter each discipline.

Matrix \(A\) is an input-output matrix associated with an economy, and matrix \(D\) (units in millions of dollars) is a demand vector. In each problem,find the final outputs of each industry such that the demands of industry and the consumer sector are met. $$ A=\left[\begin{array}{ll} 0.4 & 0.2 \\ 0.3 & 0.5 \end{array}\right] \text { and } D=\left[\begin{array}{l} 12 \\ 24 \end{array}\right] $$

The total number of passengers riding a certain city bus during the morning shift is 1000 . If the child's fare is \(\$ .50\), the adult fare is \(\$ 1.50\), and the total revenue from the fares in the morning shift is \(\$ 1300\), how many children and how many adults rode the bus during the morning shift?

For the opening night at the Opera House, a total of 1000 tickets were sold. Front orchestra seats cost $$\$ 80$$ apiece, rear orchestra seats cost $$\$ 60$$ apiece, and front balcony seats cost $$\$ 50$$ apiece. The combined number of tickets sold for the front orchestra and rear orchestra exceeded twice the number of front balcony tickets sold by 400. The total receipts for the performance were $$\$ 62,800$$. Determine how many tickets of each type were sold.
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