/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Give the augmented matrix in ech... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 16: Problem 6

Give the augmented matrix in echelon form of a system of linear equations in \(x\) and \(y .\) Find \(x\) and \(y\). $$ \left(\begin{array}{cc|c} 2 & -1 & 0.01 \\ 0 & 10 & 0.5 \end{array}\right) $$

Short Answer

Expert verified
Answer: The solution is \(x = 0.03\) and \(y = 0.05\).

Step by step solution

01

Convert the augmented matrix into linear equations

To convert the given augmented matrix back into a system of linear equations, we can simply take each row in the matrix and create a corresponding equation using the coefficients for \(x\) and \(y\) and the constants on the right side of the matrix. The given matrix: $$ \left(\begin{array}{cc|c} 2 & -1 & 0.01 \\ 0 & 10 & 0.5 \end{array}\right) $$ can be represented as the following system of linear equations: 1. \(2x - y = 0.01\) 2. \(10y = 0.5\)
02

Solve for y in the second equation

We can solve the second equation for \(y\) by simply dividing by 10: $$ y = \frac{0.5}{10} = 0.05 $$
03

Substitute the value of y to the first equation and solve for x

Now that we know the value of \(y\), we can substitute it back into the first equation and solve for \(x\): $$ 2x - 0.05 = 0.01 $$ Add 0.05 to both sides of the equation: $$ 2x = 0.06 $$ Now, divide by 2 to find the value of \(x\): $$ x = \frac{0.06}{2} = 0.03 $$
04

The solution

We have found the values for \(x\) and \(y\) that satisfy the given system of linear equations represented by the augmented matrix. The solution is \(x = 0.03\) and \(y = 0.05\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Echelon Form
Echelon form is an organized way to display the equations of a system as a matrix. Converting to echelon form simplifies working with multiple equations. To achieve this form:
  • You place zeros below the leading entries (numerical coefficients of the variables) in each growing row.
  • The focus is on the upper triangle of the matrix, making it easier to solve.
The aim is to make the matrix step-by-step closer to a simpler form. This helps in clearly identifying the solutions of the system of equations, making it insightful to analyze major characteristics of the equations involved.
System of Linear Equations
A system of linear equations consists of multiple linear equations that share common variables. The objective is to find values for these variables that satisfy all equations simultaneously. In this context, the system is represented by an augmented matrix:\[\begin{array}{cc|c} 2 & -1 & 0.01 \ 0 & 10 & 0.5 \end{array}\]

This system translates to two equations:

  • \(2x - y = 0.01\)
  • \(10y = 0.5\)
Systems of linear equations are foundational in different fields, from engineering to economics, where multiple constraints must be met. They offer a way to model relationships between quantities that are directly proportional or inversely related.
Solving Linear Equations
Solving linear equations involves finding values for the variables that make the equation true. With matrices, this process includes interpreting the rows of the matrix as separate equations and manipulating them to find solutions:**Step 1**: Translate the matrix into equations. This involves converting each row back into its corresponding linear equation form.**Step 2**: Solve for one of the variables. In our exercise, equation \( 10y = 0.5 \) directly gives us \( y = 0.05 \).**Step 3**: Substitute this value back into the first equation. It allows you to find the second variable precisely: \[ 2x - 0.05 = 0.01 \] Adding 0.05 to both sides results in: \[ 2x = 0.06 \] Dividing by 2 results in: \[ x = 0.03 \]This step-by-step process ensures accuracy and clarity, leading to the complete solution \(x = 0.03\) and \(y = 0.05\). Each step builds on the previous one, systematically unraveling the values needed to resolve the system effectively.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

2\. A rental company has 350 cars. Among these cars \(D\) of them are at its downtown location, \(P\) of them at the port and \(A\) of them at the airport. Each year, \(10 \%\) of the cars rented downtown are returned at the port, \(20 \%\) of the cars rented at the port are returned at the airport, and \(5 \%\) of the cars rented at the airport are returned downtown. All other cars are returned to where they were rented. The number of cars at each location remains the same each year. (a) Write four equations from the given information. (b) Write an augmented matrix for this system of equations. (c) Use row operations to find the number of cars at each location.

Find a single vector resulting from the operations. \(\left(\begin{array}{cc}5 & 10 \\ 20 & 40\end{array}\right)\left(\begin{array}{l}3 \\ 7\end{array}\right)\)

A Double-Double fast food diet has approximately double fat to protein and double carbohydrates to fat. One day, you eat \(B\) burgers, \(S\) shakes, and \(F\) servings of fries, for a total of \(125 \mathrm{gm}\) of protein, \(250 \mathrm{gm}\) of fat, and \(505 \mathrm{gm}\) of carbohydrates. Table 16.2 shows the nutritional information for three fast foods. $$ \begin{aligned} &\text { Table } 16.2\\\ &\begin{array}{c|c|c|c} \hline & \text { Burger } & \text { Shake } & \text { Fries } \\ \hline \text { Fat }(\mathrm{gm}) & 40 & 30 & 25 \\ \hline \text { Carbohydrates }(\mathrm{gm}) & 50 & 115 & 60 \\ \hline \text { Protein }(\mathrm{gm}) & 30 & 15 & 5 \\ \hline \end{array} \end{aligned} $$ (a) Write a system of equations describing this situation. (b) Write an augmented matrix for the system. (c) Use row operations to find how many of each item you eat.

In Problems \(24-25,\) refer to \(\mathbf{R}\) and \(\mathbf{M},\) matrices of mean SAT scores. The columns are mean SAT reasoning scores for the years \(2001-2008 .\) The first row is scores for males and the second row is scores for females. Matrix \(\mathbf{R}\) is the Critical Reading scores, and matrix \(\mathbf{M}\) is the Mathematics scores. \(^{1}\). \(\mathbf{R}=\left(\begin{array}{llllllll}509 & 507 & 512 & 512 & 513 & 505 & 504 & 504 \\ 502 & 502 & 503 & 504 & 505 & 502 & 502 & 500\end{array}\right)\) \(\mathbf{M}=\left(\begin{array}{cccccccc}533 & 534 & 537 & 537 & 538 & 536 & 533 & 533 \\ 498 & 500 & 503 & 501 & 504 & 502 & 499 & 500\end{array}\right)\).Calculate \(\mathbf{M}-\mathbf{R}\). What does this represent?

The augmented matrices for a system of equations have been reduced to a form which makes them easy to solve. Find the values of \(x\) and \(y\) by inspection (that is, without writing anything down except the answer). $$ \left(\begin{array}{ll|l} 1 & 0 & 2 \\ 0 & 1 & 3 \end{array}\right) $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.