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Chapter 7: Problem 38

For the following exercises, solve each system by Gaussian elimination. $$ \begin{array}{l}{0.2 x+0.1 y-0.3 z=0.2} \\ {0.8 x+0.4 y-1.2 z=0.1} \\ {1.6 x+0.8 y-2.4 z=0.2}\end{array} $$

Short Answer

Expert verified
The system is inconsistent and has no solutions.

Step by step solution

01

Set Up the Augmented Matrix

Write the systems of equations as an augmented matrix. For the given system, the augmented matrix is: \[\begin{bmatrix}0.2 & 0.1 & -0.3 & | & 0.2\0.8 & 0.4 & -1.2 & | & 0.1\1.6 & 0.8 & -2.4 & | & 0.2\end{bmatrix}\]
02

Scale Row 1

To make calculations easier, scale the first row by multiplying it by 5 to avoid decimals:\[\begin{bmatrix}1 & 0.5 & -1.5 & | & 1\0.8 & 0.4 & -1.2 & | & 0.1\1.6 & 0.8 & -2.4 & | & 0.2\end{bmatrix}\]
03

Eliminate x from Row 2

Subtract 0.8 times the new Row 1 from Row 2 to eliminate the x-term:\[Row2 = Row2 - 0.8 \times Row1\]The augmented matrix becomes:\[\begin{bmatrix}1 & 0.5 & -1.5 & | & 1\0 & 0 & 0 & | & -0.7\1.6 & 0.8 & -2.4 & | & 0.2\end{bmatrix}\]
04

Eliminate x from Row 3

Subtract 1.6 times the new Row 1 from Row 3 to eliminate the x-term:\[Row3 = Row3 - 1.6 \times Row1\]The augmented matrix becomes:\[\begin{bmatrix}1 & 0.5 & -1.5 & | & 1\0 & 0 & 0 & | & -0.7\0 & 0 & 0 & | & -1.4\end{bmatrix}\]
05

Analyze the System

Notice that both Row 2 and Row 3 have all-zero coefficients for x, y, and z, but different constants in the augmented part. This indicates a contradiction (inconsistent system), as they imply the equations 0 = -0.7 and 0 = -1.4, which are impossible.

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Key Concepts

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

Understanding the Augmented Matrix
In solving systems of equations like the one given, it's common to use an augmented matrix. This is a way of organizing coefficients and constants into a single matrix for easier manipulation. The augmented matrix combines all the equations' coefficients into rows, and the constants on the right side of the equations form the last column.

Simply put, each row of the augmented matrix represents one equation.
  • The coefficients of variables (e.g., numbers before \(x\), \(y\), \(z\)) are arranged in the left columns.
  • The constants (the numbers alone on the right of the equal sign) are the final column.
The vertical line separating the last column from the others simply divides the coefficients from the constants. In essence, the augmented matrix is a tool that simplifies row operations and helps us to solve systems effectively.
Identifying an Inconsistent System
An inconsistent system occurs when there is no possible set of solutions that satisfies all the equations simultaneously. In this context, when reducing the augmented matrix using Gaussian elimination, an inconsistent system will show up as a row where the left-hand side equals zero, but the right-hand side does not.

This contradiction means there鈥檚 no way the equation can be true. For example:
  • If during matrix reduction, you end up with a row such as \(0x + 0y + 0z = -0.7\), the equation says \(0 = -0.7\), which is impossible.
  • Similarly, another row such as \(0x + 0y + 0z = -1.4\) implies \(0 = -1.4\), further confirming inconsistency.
Such rows highlight the inconsistency and lead us to conclude that the original set of equations cannot be solved.
How to Eliminate Variables
Eliminating variables is a crucial part of Gaussian elimination. This process simplifies the system of equations by making some variables disappear from certain equations, usually through strategic subtraction or addition.
  • The main idea is to transform the system into a triangular form, where the first equation deals with three variables, the second with two, and the last with one.
  • In practice, you perform row operations on the matrix, aiming to get zeros in strategic places.
  • For instance, to eliminate \(x\) from Row 2, you can subtract a multiple of Row 1 after scaling appropriately to match the coefficient of \(x\) in Row 2.
  • This operation involves finding a suitable factor (like 0.8 in the example), multiplying Row 1 by it, and subtracting from Row 2.
By doing so, you systematically remove variables and make the matrix easier to solve, reducing complexity step by step.

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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 market, the three most popular vegetables make up 53\(\%\) of vegetable sales. Corn has 4\(\%\) higher sales than broccoli, which has 5\(\%\) more sales than onions. What percentage does each vegetable have in the market share?

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{aligned} 13 x-17 y+16 z &=73 \\ -11 x+15 y+17 z &=61 \\ 46 x+10 y-30 z &=-18 \end{aligned} $$

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. The three most popular ice cream flavors are chocolate, strawberry, and vanilla, comprising \(83 \%\) of the flavors sold at an ice cream shop. If vanilla sells \(1 \%\) more than twice strawberry, and chocolate sells \(11 \%\) more than vanilla, how much of the total ice cream consumption are the vanilla, chocolate, and strawberry flavors?

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{aligned} -5 x+2 y-4 z &=-47 \\ 4 x-3 y-z &=-94 \\ 3 x-3 y+2 z &=94 \end{aligned} $$

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. You decide to paint your kitchen green. You create the color of paint by mixing yellow and blue paints. You cannot remember how many gallons of each color went into your mix, but you know there were 10 gal total. Additionally, you kept your receipt, and know the total amount spent was \(29.50. If each gallon of yellow costs \)2.59, and each gallon of blue costs $3.19, how many gallons of each color go into your green mix?
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