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Chapter 12: Problem 32

The reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use \(x, y ;\) or \(x, y, z ;\) or \(x_{1}, x_{2}, x_{3}, x_{4}\) as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution. $$ \left[\begin{array}{lll|l} 1 & 0 & 4 & 4 \\ 0 & 1 & 3 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right] $$

Short Answer

Expert verified
The system is consistent. The general solution is \(x = 4 - 4z, y = 2 - 3z, z = z\).

Step by step solution

01

Write the System of Equations

Translate each row of the augmented matrix back into a linear equation. The matrix \[ \begin{bmatrix} 1 & 0 & 4 & 4 \ 0 & 1 & 3 & 2 \ 0 & 0 & 0 & 0 \ \end{bmatrix} \] corresponds to the following system of equations: \[ \begin{cases} x + 4z = 4 \ y + 3z = 2 \end{cases} \]
02

Determine Consistency

Check each row of the matrix to see if a contradiction exists. Since the third row is all zeros, which equates to the equation \(0 = 0\), the system has no contradiction. Therefore, it is consistent.
03

Solve the System

Express the solutions in terms of the free variable, which here is \(z\). From the first equation, solve for \(x\): \[ x = 4 - 4z \] From the second equation, solve for \(y\): \[ y = 2 - 3z \]
04

Final Solution

The solution to the system is: \[ (x, y, z) = (4 - 4z, 2 - 3z, z) \] where \(z\) can be any real number.

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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** is a collection of two or more linear equations involving the same set of variables. For example, in our exercise, the set of linear equations derived from the matrix are:
  • x + 4z = 4
  • y + 3z = 2
Each equation represents a line, plane, or hyperplane depending on the number of variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. For our exercise, we need to determine the values of x, y, and z such that both equations are true at the same time.
Consistency of a System
The **consistency of a system** refers to whether a system of equations has at least one solution. A system is:
  • **Consistent:** If there is one or more solutions. This happens when there are no contradictions within the equations.
  • **Inconsistent:** If there are no solutions. This happens when the equations contradict each other.
In our exercise, the reduced row echelon form (RREF) matrix: does not show any contradictions like \(0 = 1\), which means the system is consistent. The third row consisting of all zeros signifies that there is no new information from that row. Thus, the system of equations has solutions.
Solution of System of Equations
The **solution of a system of equations** consists of all variable values that satisfy every equation in the system. In our exercise:
  • From the first equation, x + 4z = 4, solve for x: \[ x = 4 - 4z \]
  • From the second equation, y + 3z = 2, solve for y: \[ y = 2 - 3z \]
Combining these results, the solution can be represented as: \[ (x, y, z) = (4 - 4z, 2 - 3z, z) \]This notation shows that the solution depends on the value of z. We can use this result to determine the values of x and y for any real number z.
Free Variable
A **free variable** in a system of equations is a variable that can take on any value. Its presence often indicates there are infinitely many solutions.
In our exercise, z is a free variable because it does not have a unique solution 鈥� any value of z will satisfy the system of equations.
This means the solution can be written where x and y are expressed in terms of z:
  • x = 4 - 4z
  • y = 2 - 3z
Since z can be any real number, we have an infinite set of solutions characterized by the parameter z. Thus, z being a free variable allows flexibility in the solutions.

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Most popular questions from this chapter

A Florida juice company completes the preparation of its products by sterilizing, filling, and labeling bottles. Each case of orange juice requires 9 minutes (min) for sterilizing, 6 min for filling, and 1 min for labeling. Each case of grapefruit juice requires 10 min for sterilizing, 4 min for filling, and 2 min for labeling. Each case of tomato juice requires 12 min for sterilizing, 4 min for filling, and 1 min for labeling. If the company runs the sterilizing machine for 398 min, the filling machine for 164 min, and the labeling machine for 58 min, how many cases of each type of juice are prepared?

IS-LM Model in Economics In economics, the IS curve is a linear equation that represents all combinations of income \(Y\) and interest rates \(r\) that maintain an equilibrium in the market for goods in the economy. The LM curve is a linear equation that represents all combinations of income \(Y\) and interest rates \(r\) that maintain an equilibrium in the market for money in the economy. In an economy, suppose that the equilibrium level of income (in millions of dollars) and interest rates satisfy the system of equations $$ \left\\{\begin{array}{l} 0.06 Y-5000 r=240 \\ 0.06 Y+6000 r=900 \end{array}\right. $$ Find the equilibrium level of income and interest rates.

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{aligned} x+y &=9 \\ 2 x &-z=13 \\ 3 y+2 z &=7 \end{aligned}\right. $$

Solve: \(\frac{5 x}{x+2}=\frac{x}{x-2}\)

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. What is the amount that results if \(\$ 2700\) is invested at \(3.6 \%\) compounded monthly for 3 years?
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