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Chapter 8: Problem 17

In Exercises \(9-26\), put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent. $$ \left\\{\begin{aligned} x-y+z &=-4 \\ -3 x+2 y+4 z &=-5 \\ x-5 y+2 z &=-18 \end{aligned}\right. $$

Short Answer

Expert verified
The system is consistent independent with solution: \(x = -\frac{5}{3}\), \(y = \frac{35}{9}\), \(z = \frac{14}{9}\).

Step by step solution

01

Organize the System

The given system of linear equations is: \(x - y + z = -4\), \(-3x + 2y + 4z = -5\), and \(x - 5y + 2z = -18\). To solve this, we will sequentially eliminate variables to convert the system into triangular form.
02

Eliminate x from Second and Third Equations

To create the upper triangular form, eliminate \(x\) from the second and third equations. Start by subtracting the first equation from itself multiplied by 3 from the second equation: - Equation 2: \([-3x + 2y + 4z = -5]\); Add \([3(x - y + z = -4)]\).This yields: \(-3x + 2y + 4z + 3x - 3y + 3z = -5 + 12\)\(-y + 7z = 7\)This is now the new Equation 2.For the third equation, subtract the first equation: - Equation 3: \([x - 5y + 2z = -18]\); Add \([-1(x - y + z = -4)]\).This yields:\(x - 5y + 2z - x + y - z = -18 + 4\)\(-4y + z = -14\)Now the new Equation 3 is \(-4y + z = -14\).
03

Convert to Triangular Form

The modified system is now:1. \(x - y + z = -4\)2. \(-y + 7z = 7\)3. \(-4y + z = -14\)In these equations, Equation 1 is fine; terms in Equation 2 and Equation 3 correctly lined up to isolate \(y\) and \(z\). The system is now in a form where we can solve for the remaining values sequentially.
04

Solve for z from Third Equation

Solve for \(z\) from the third equation: \(-4y + z = -14\)By rearranging, we find:\(z = 4y - 14\).Substitute this expression for \(z\) in the second equation to solve for \(y\).
05

Substitute z Expression into Second Equation

Substitute \(z = 4y - 14\) into \(-y + 7z = 7\):\(-y + 7(4y - 14) = 7\)\(-y + 28y - 98 = 7\)\(27y - 98 = 7\)Add 98 on both sides:\(27y = 105\)Divide by 27:\(y = \frac{105}{27} = \frac{35}{9}\)
06

Solve for z

Substitute \(y = \frac{35}{9}\) back into the equation for \(z\):\(z = 4y - 14 = 4(\frac{35}{9}) - 14\)\(z = \frac{140}{9} - \frac{126}{9}\)\(z = \frac{14}{9}\)
07

Solve for x

Substitute \(y = \frac{35}{9}\) and \(z = \frac{14}{9}\) back into the first equation:\(x - y + z = -4\)\(x - \frac{35}{9} + \frac{14}{9} = -4\)\(x - \frac{21}{9} = -4\)\(x = -4 + \frac{21}{9}\)\(x = -\frac{36}{9} + \frac{21}{9}\)\(x = -\frac{15}{9} = -\frac{5}{3}\)
08

Classify the System

The system has a unique solution: \(x = -\frac{5}{3}\), \(y = \frac{35}{9}\), \(z = \frac{14}{9}\). Therefore, it is a consistent independent system.

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Key Concepts

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

Systems of Linear Equations
A system of linear equations consists of two or more equations with the same set of variables. The objective is to find the values of these variables that simultaneously satisfy all the equations. Linear equations are equations of the first degree, meaning they have variables raised only to the power of one. Commonly seen in algebra, these systems can model various real-world problems, such as determining the intersection of lines or optimizing production conditions in business. Understanding the concept of systems of linear equations can help you recognize both solvable (consistent) and unsolvable (inconsistent) scenarios. Solving such systems involves finding a common solution for all equations.
Consistent Independent System
When dealing with systems of linear equations, a consistent independent system is one that has exactly one solution. This means that the equations in the system intersect at a single point on a graph. Typically, for two variables, the lines will cross at one distinct point. This unique intersection indicates there is one set of variable values that satisfies all equations in the system. Such systems are rightly named 'consistent' because they have a solution, and 'independent' because each equation contributes unique information to determine the solution. Determining this allows us to understand the uniqueness of the system's solution.
Solving Equations
The process of solving equations involves finding the values of variables that make the equations true. In a system of equations, this process often requires manipulation of the equations to isolate variables one at a time. A common approach is to transform equations to isolate one variable in terms of others, often using substitution or elimination methods.
  • Substitution: Solve one equation for one variable, then substitute that expression in other equations.
  • Elimination: Add or subtract equations to eliminate one variable, simplifying the process of finding solutions.
Once one variable is found, it is substituted back into original equations to find the values of remaining variables. This sequential process ultimately unveils the complete solution set for the system.
Linear Algebra Concepts
Linear algebra encompasses many concepts that help in understanding and solving systems of linear equations. At the core is the concept of vectors and matrices, which enable compact representations of linear systems. Triangular form, specifically upper triangular form, simplifies systems to allow step-by-step solving. Other important concepts include determinants and inverse matrices, which aid in solving larger systems. Each of these concepts plays a crucial role in breaking down complex problems into manageable steps and finding solutions efficiently. Understanding these principles is essential for anyone aiming to master higher-level mathematics or applied fields like physics or economics.

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Most popular questions from this chapter

At The Old Home Fill'er Up and Keep on a-Truckin' Cafe, Mavis mixes two different types of coffee beans to produce a house blend. The first type costs \(\$ 3\) per pound and the second costs \(\$ 8\) per pound. How much of each type does Mavis use to make 50 pounds of a blend which costs \(\$ 6\) per pound?

Let \(z=a+b i\) and \(w=c+d i\) be arbitrary complex numbers. Associate \(z\) and \(w\) with the matrices $$ Z=\left[\begin{array}{rr} a & b \\ -b & a \end{array}\right] \text { and } W=\left[\begin{array}{rr} c & d \\ -d & c \end{array}\right] $$ Show that complex number addition, subtraction and multiplication are mirrored by the associated matrix arithmetic. That is, show that \(Z+W, Z-W\) and \(Z W\) produce matrices which can be associated with the complex numbers \(z+w, z-w\) and \(z w,\) respectively.

In Exercises 7 - 18 , find the partial fraction decomposition of the following rational expressions. $$ \frac{-2 x^{2}+20 x-68}{x^{3}+4 x^{2}+4 x+16} $$

In Exercises \(23-29,\) consider the following scenario. In the small village of Pedimaxus in the country of Sasquatchia, all 150 residents get one of the two local newspapers. Market research has shown that in any given week, \(90 \%\) of those who subscribe to the Pedimaxus Tribune want to keep getting it, but \(10 \%\) want to switch to the Sasquatchia Picayune. Of those who receive the Picayune, \(80 \%\) want to continue with it and \(20 \%\) want switch to the Tribune. We can express this situation using matrices. Specifically, let \(X\) be the 'state matrix' given by $$ X=\left[\begin{array}{l} T \\ P \end{array}\right] $$ where \(T\) is the number of people who get the Tribune and \(P\) is the number of people who get the Picayune in a given week. Let \(Q\) be the 'transition matrix' given by $$ Q=\left[\begin{array}{ll} 0.90 & 0.20 \\ 0.10 & 0.80 \end{array}\right] $$ such that \(Q X\) will be the state matrix for the next week. Let's assume that when Pedimaxus was founded, all 150 residents got the Tribune. (Let's call this Week \(0 .)\) This would mean $$ X=\left[\begin{array}{r} 150 \\ 0 \end{array}\right] $$ Since \(10 \%\) of that 150 want to switch to the Picayune, we should have that for Week 1, 135 people get the Tribune and 15 people get the Picayune. Show that \(Q X\) in this situation is indeed $$ Q X=\left[\begin{array}{r} 135 \\ 15 \end{array}\right] $$

A local buffet charges \(\$ 7.50\) per person for the basie buffet and \(\$ 9.25\) for the deluxe buffet (which includes crab legs.) If 27 diners went out to eat and the total bill was \(\$ 227.00\) before taxes, how many chose the basic buffet and how many chose the deluxe buffet?
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