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Chapter 2: Problem 37

Solve the system of linear equations using the Gauss-Jordan elimination method. $$ \begin{array}{l} 2 x-3 y=-8 \\ 4 x+y=-2 \end{array} $$

Short Answer

Expert verified
The solution to the given system of linear equations using Gauss-Jordan elimination is: \(x = -1\) and \(y = 2\).

Step by step solution

01

Write the augmented matrix

First, we represent the system of linear equations as an augmented matrix. For the given system: $$ \begin{array}{l} 2 x-3 y=-8 \\ 4 x+y=-2 \end{array} $$ The augmented matrix is: $$ \left[\begin{array}{cc|c} 2 & -3 & -8 \\ 4 & 1 & -2 \end{array}\right] $$
02

Perform Gauss-Jordan elimination

We must perform row operations on the matrix to achieve the RREF. We want to have a 1 in the top left corner, and 0s below it. We can accomplish this by following these sequence of operations: 1. Divide row 1 by 2: $ \left[\begin{array}{cc|c} 1 & -\frac{3}{2} & -4 \\ 4 & 1 & -2 \end{array}\right] $ 2. Replace row 2 with row 2 - 4 * row 1: $ \left[\begin{array}{cc|c} 1 & -\frac{3}{2} & -4 \\ 0 & 7 & 14 \end{array}\right] $ 3. Divide row 2 by 7: $ \left[\begin{array}{cc|c} 1 & -\frac{3}{2} & -4 \\ 0 & 1 & 2 \end{array}\right] $ 4. Replace row 1 with row 1 + \(\frac{3}{2}\) * row 2: $ \left[\begin{array}{cc|c} 1 & 0 & -1 \\ 0 & 1 & 2 \end{array}\right] $ Now, our matrix is in RREF, and we can interpret the results.
03

Extract the solution from the RREF

The RREF matrix corresponds to the following system of equations: $$ \begin{array}{l} x = -1 \\ y = 2 \end{array} $$ So, the solution to the system of linear equations is: \(x = -1\) and \(y = 2\).

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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 consists of two or more linear equations involving the same set of variables. In our exercise, we have the system:
  • 2x - 3y = -8
  • 4x + y = -2
Each equation represents a line in a two-dimensional plane. The goal is to find the point(s) where these lines intersect, representing the solutions, or values of x and y, that satisfy both equations.
Solving a system can be visualized as finding the meeting point of the lines. Mathematically, this involves determining values for x and y that work for both equations.
Augmented Matrix
To apply linear algebra techniques like Gauss-Jordan elimination, we convert a system of linear equations into an augmented matrix. An augmented matrix combines the coefficients of the variables and the constants from the equations into a single matrix. For the given system:\[\begin{array}{cc|c}2 & -3 & -8 \4 & 1 & -2\end{array}\]The columns before the vertical line correspond to the coefficients of x and y, while the column after is the constants from the equations.
A matrix is a compact way of handling systems of equations, allowing us to perform operations systematically to find solutions.
Row Operations
Row operations are key to transforming the augmented matrix into a more useful form, called the reduced row-echelon form (RREF). These operations include:
  • Swapping two rows (Row interchange)
  • Multiplying a row by a non-zero scalar (Row scaling)
  • Adding or subtracting multiples of one row to another row (Row replacement)
By systematically applying these, we can simplify the matrix without changing the solutions of the equations. The main objective is to form a diagonal of 1s with zeros elsewhere, simplifying the matrix to a state that easily reveals the solution.
Reduced Row-Echelon Form (RREF)
The reduced row-echelon form is the final simplified form of a matrix, achieved through row operations. It satisfies these conditions:
  • Each leading entry in a row is 1.
  • Each leading 1 is the only non-zero entry in its column.
  • The leading 1 in a row appears to the right of the leading 1 in the previous row.
  • Rows with all zeros are at the bottom.
In the context of our exercise, the RREF of the matrix:\[\begin{array}{cc|c}1 & 0 & -1 \0 & 1 & 2\end{array}\]This matrix straightforwardly leads us to the solution: \(x = -1\) and \(y = 2\). RREF makes it easy to read off the values of x and y directly, as each variable corresponds to its own row.

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Most popular questions from this chapter

Hartman Lumber Company has two branches in the city. The sales of four of its products for the last year (in thousands of dollars) are represented by the matrix $$ B=\begin{array}{l} \text { Branch I } \\ \text { Branch II } \end{array} \quad\left[\begin{array}{rrrr} 5 & 3 & C & D \\ 3 & 4 & 6 & 8 \end{array}\right] $$ For the present year, management has projected that the sales of the four products in branch I will be \(10 \%\) more than the corresponding sales for last year and the sales of the four products in branch II will be \(15 \%\) more than the corresponding sales for last year. a. Show that the sales of the four products in the two branches for the current year are given by the matrix \(A B\), where $$ A=\left[\begin{array}{ll} 1.1 & 0 \\ 0 & 1.15 \end{array}\right] $$ Compute \(A B\). b. Hartman has \(m\) branches nationwide, and the sales of \(n\) of its products (in thousands of dollars) last year are represented by the matrix Product $$ \left.B=\begin{array}{c} 1 & 2 & 3 & \cdots & n \\ \text { Branch } 1 \\ \text { Branch 2 } \\ \vdots & a_{11} & a_{12} & a_{13} & \cdots & a_{1 n} \\ a_{21} & a_{22} & a_{23} & \cdots & a_{2 n} \\ \vdots & \vdots & \vdots & & \vdots \\ a_{m 1} & a_{m 2} & a_{m 3} & \cdots & a_{m n} \end{array}\right] $$ Also, management has projected that the sales of the \(n\) products in branch 1 , branch \(2, \ldots\), branch \(m\) will be \(r_{1} \%, r_{2} \%, \ldots, r_{m} \%\), respectively, more than the corresponding sales for last year. Write the matrix \(A\) such that \(A B\) gives the sales of the \(n\) products in the \(m\) branches for the current year.

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.

Solve the system of linear equations using the Gauss-Jordan elimination method. $$ \begin{array}{r} x-2 y=8 \\ 3 x+4 y=4 \end{array} $$

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. Michael Perez has a total of $$\$ 2000$$ on deposit with two savings institutions. One pays interest at the rate of \(6 \%\) lyear, whereas the other pays interest at the rate of \(8 \% /\) year. If Michael earned a total of $$\$ 144$$ in interest during a single year, how much does he have on deposit in each institution?

The total output of loudspeaker systems of the Acrosonic Company at their three production facilities for May and June is given by the matrices \(A\) and \(B\), respectively, where $$ \begin{array}{l} \left.\begin{array}{ccccc} & \text { Model } & \text { Model } & \text { Model } & \text { Model } \\ & \text { A } & \text { B } & \text { C } & \text { D } \\ \text { Location I } & {[320} & 280 & 460 & 280 \\ A= & \text { Location II } & 480 & 360 & 580 & 0 \\ \text { Location III } & 540 & 420 & 200 & 880 \end{array}\right]\\\ \left.\begin{array}{ccccc} & \text { Model } & \text { Model } & \text { Model } & \text { Model } \\ & \text { A } & \text { B } & \text { C } & \text { D } \\ \text { Location I } & 210 & 180 & 330 & 180 \\ B=\text { Location II } & 400 & 300 & 450 & 40 \\ \text { Location III } & 420 & 280 & 180 & 740 \end{array}\right] \end{array} $$ The unit production costs and selling prices for these loudspeakers are given by matrices \(C\) and \(D\), respectively, where \(\begin{array}{c}\text { Model A } \\ C=\begin{array}{l}\text { Model B } \\\ \text { Model C } \\ \text { Model D }\end{array} & {\left[\begin{array}{l}120 \\ 180 \\ 260 \\ 500\end{array}\right] \text { and } \quad D=\begin{array}{c}\text { Model A } \\ \text { Model B } \\\ \text { Model C } \\ \text { Model D }\end{array}} & {\left[\begin{array}{l}160 \\ 250 \\ 350 \\\ 700\end{array}\right]}\end{array}\) Compute the following matrices and explain the meaning of the entries in each matrix. a. \(A C\) b. \(A D\) c. \(B C\) d. \(B D\) e. \((A+B) C\) f. \((A+B) D\) g. \(A(D-C)\) h. \(B(D-C)\) i. \((A+B)(D-C)\)
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