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Chapter 7: Problem 13

Write the linear system from the augmented matrix. \(\left[\begin{array}{rrr|r}3 & 2 & 0 & 3 \\ -1 & -9 & 4 & -1 \\ 8 & 5 & 7 & 8\end{array}\right]\)

Short Answer

Expert verified
3x_1 + 2x_2 = 3, -x_1 - 9x_2 + 4x_3 = -1, 8x_1 + 5x_2 + 7x_3 = 8.

Step by step solution

01

Understand the Matrix Format

Identify the given augmented matrix \[\left[\begin{array}{rrr|r} 3 & 2 & 0 & 3 \ -1 & -9 & 4 & -1 \ 8 & 5 & 7 & 8 \end{array}\right]\]This represents a system of three equations. Each row corresponds to an equation, and each column (before the vertical bar) corresponds to coefficients of the variables.
02

Assign Variables

Let's denote the variables as follows:- Let \(x_1\) correspond to the first column of coefficients.- Let \(x_2\) correspond to the second column of coefficients.- Let \(x_3\) correspond to the third column of coefficients.The column after the vertical bar gives the constants on the right-hand side of each equation.
03

Write Equations from Rows

For each row of the matrix, write an equation using the coefficients and the constants.**First row:**The equation is:\[ 3x_1 + 2x_2 + 0x_3 = 3 \]**Second row:**The equation is:\[ -1x_1 - 9x_2 + 4x_3 = -1 \]**Third row:**The equation is:\[ 8x_1 + 5x_2 + 7x_3 = 8 \]
04

Present the Linear System

The linear system of equations corresponding to the augmented matrix is:\[\begin{align*}3x_1 + 2x_2 &= 3 \-x_1 - 9x_2 + 4x_3 &= -1 \8x_1 + 5x_2 + 7x_3 &= 8\end{align*}\]

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Key Concepts

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

Augmented Matrix
An augmented matrix is a powerful tool used in linear algebra to represent systems of linear equations. It is essentially a compact way of writing down the coefficients and constants of these equations in a matrix form.
In an augmented matrix, each row corresponds to an equation in the system while each column represents the coefficients of a specific variable. The last column, separated by a vertical line, contains the constants from the equations鈥� right-hand sides.
For example, the augmented matrix \[\left[\begin{array}{rrr|r} 3 & 2 & 0 & 3 \ -1 & -9 & 4 & -1 \ 8 & 5 & 7 & 8\end{array}\right]\] corresponds to a system of three equations with three variables.
Linear Equations
Linear equations are mathematical statements of equality involving linear expressions. They form straight lines when graphed on a coordinate plane and contain no exponents or powers greater than one.
A system of linear equations involves multiple such equations, which can be solved simultaneously to find the values of the variables involved.
From the augmented matrix, you can express the system of linear equations as follows:
  • The first row translates to \(3x_1 + 2x_2 + 0x_3 = 3\).
  • The second row translates to \(-1x_1 - 9x_2 + 4x_3 = -1\).
  • The third row translates to \(8x_1 + 5x_2 + 7x_3 = 8\).
These equations are linear because each term is either a constant or a constant times a variable.
Variables and Coefficients
In the context of linear equations, variables are the unknowns that we aim to solve for, like \(x_1\), \(x_2\), and \(x_3\) in our example. Coefficients are the numbers that multiply the variables in each term of the equation.
In our augmented matrix, variables are assigned to specific columns:
  • \(x_1\) corresponds to the first column of coefficients.
  • \(x_2\) corresponds to the second column of coefficients.
  • \(x_3\) corresponds to the third column of coefficients.
Learning to identify the variables and their respective coefficients is essential in setting up the correct equations from an augmented matrix, helping solve the system effectively.

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Most popular questions from this chapter

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer鈥檚 Rule. At a women鈥檚 prison down the road, the total number of inmates aged 20鈥�49 totaled 5,525. This year, the 20鈥�29 age group increased by 10%, the 30鈥�39 age group decreased by 20%, and the 40鈥�49 age group doubled. There are now 6,040 prisoners. Originally, there were 500 more in the 30鈥�39 age group than the 20鈥�29 age group. Determine the prison population for each age group last year.

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{aligned} 4 x-6 y+8 z &=10 \\ -2 x+3 y-4 z &=-5 \\ 12 x+18 y-24 z &=-30 \end{aligned} $$

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. Bikes'R'Us manufactures bikes, which sell for \(\$ 250\). It costs the manufacturer \(\$ 180\) per bike, plus a startup fee of \(\$ 3,500\). After how many bikes sold will the manufacturer break even?

For the following exercises, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix. A food drive collected two different types of canned goods, green beans and kidney beans. The total number of collected cans was 350 and the total weight of all donated food was 348 lb, 12 oz. If the green bean cans weigh 2 oz less than the kidney bean cans, how many of each can was donated?

For the following exercises, use the determinant function on a graphing utility. \(\left|\begin{array}{llll}1 & 0 & 8 & 9 \\ 0 & 2 & 1 & 0 \\ 1 & 0 & 3 & 0 \\\ 0 & 2 & 4 & 3\end{array}\right|\)
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