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Chapter 2: Problem 64

Give an example of a system of three linear equations with the general solution \(x=y-1, y\) arbitrary, \(z=y\). (Check your system by solving it.)

Short Answer

Expert verified
The system of linear equations with the given general solution \(x = y - 1\), \(y\) arbitrary, and \(z = y\) is: \(\begin{cases} x - y = -1 \\ y=y \\ z - y = 0 \end{cases}\) After checking by solving, we confirmed that the solution matches the given general solution.

Step by step solution

01

Use the given general solution to form equations

From the general solution, we have: 1. \(x = y-1\) 2. \(y\) (arbitrary) 3. \(z = y\)
02

Create a system of linear equations

Let's use the given relations between the variables to create a system of linear equations. Based on the general solution, we can create the following three equations: 1. \(x - y = -1\) 2. \(y = y\) 3. \(z - y = 0\) Now, we have obtained the system of linear equations as: \(\begin{cases} x - y = -1 \\ y=y \\ z - y = 0 \end{cases}\)
03

Check the system by solving it

Let's now solve this system of linear equations to check if its solution matches the given general solution. First, we can ignore the middle equation as \(y=y\) means \(y\) can take any value, and it is arbitrary. Then, we have the following set of equations to solve: \(\begin{cases} x - y = -1 \\ z - y = 0 \end{cases}\) From the second equation, we can isolate \(y\): \(y = z\). Now, replace the value of \(y\) from this equation into the first equation: \(x - z = -1\) After this, simply add \(z\) to both sides: \(x = z -1\) Since in the second equation, we have \(z = y\), we can replace \(z\) with \(y\): \(x = y - 1\) The system solution is \(x = y - 1, y\) arbitrary, and \(z = y\), which matches the given general solution.

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Key Concepts

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

Systems of Equations
In mathematics, a **system of equations** consists of two or more equations with the same set of variables. Solving these systems means finding values for the variables that satisfy all equations simultaneously.
There are different types of systems:
  • Consistent systems: These have at least one solution. They can be independent (one unique solution) or dependent (infinitely many solutions).
  • Inconsistent systems: These have no solutions since the equations contradict each other.
In the given problem, we have a system of three linear equations that are consistent and dependent, since they result in infinitely many solutions. This is because one variable, \(y\), is arbitrary. By finding a common set of values, like in our example, we derive the relationships between the variables to understand how they interact.
General Solution
A **general solution** expresses the solution to a system of equations in terms of one or more arbitrary parameters or variables.
These arbitrary variables indicate that there are infinitely many solutions. In our example, the arbitrary variable is \(y\), and the general solution is given by the formulas \(x = y - 1\), \(y\), and \(z = y\).
The general solution showcases how each variable relates to each other. For example, altering \(y\) changes both \(x\) and \(z\) consistently in this scenario. Therefore, our general solutions are essentially a way of expressing all potential solutions within defined limits.
Solving Linear Systems
**Solving linear systems** involves finding the values that satisfy all the equations simultaneously. There are several methods to achieve this:
  • Substitution: Solve one equation for one variable and substitute this expression into the other equations.
  • Elimination: Combine the equations in such a way that one variable cancels out, simplifying the system to fewer equations.
  • Matrix methods (like Gaussian elimination): Use matrices to perform operations and solve the system, particularly useful for larger equations.
In the example provided, substitution was the primary method used. By expressing \(z\) in terms of \(y\), we could substitute back into the other equations, illustrating how variables interconnect. This helped complete solving the system, aligning with the initial general solution given. Recognizing the relationships and simplifying the equations are key steps in solving linear systems efficiently.

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Most popular questions from this chapter

Use Gauss-Jordan row reduction to solve the given systems of equation. We suggest doing some by hand, and others using technology. HINT [See Examples 1-6.] $$ \begin{array}{r} -\frac{1}{2} x+y-\frac{1}{2} z=0 \\ -\frac{1}{2} x-\frac{1}{2} y+z=0 \\ x-\frac{1}{2} y-\frac{1}{2} z=0 \end{array} $$

Use Gauss-Jordan row reduction to solve the given systems of equation. We suggest doing some by hand, and others using technology. HINT [See Examples 1-6.] $$ \begin{aligned} x+y-& z=-2 \\ x-y-& 7 z=0 \\ 0.75 x-0.5 y+0.25 z &=14 \\ x+\quad y+\quad z &=4 \end{aligned} $$

The Fancy French Perfume Company recently had its secret formula divulged. It turned out that it was using, as the three ingredients, rose oil, oil of fermented prunes, and alcohol. Moreover, each 22 -ounce econosize bottle contained 4 more ounces of alcohol than oil of fermented prunes, while the amount of alcohol was equal to the combined volume of the other two ingredients. How much of each ingredient did it use in an econo-size bottle? \(^{14}\)

In the matrix of a system of linear equations, suppose that two of the rows are equal. What can you say about the rowreduced form of the matrix?

Big Red Bookstore wants to ship books from its warehouses in Brooklyn and Queens to its stores, one on Long Island and one in Manhattan. Its warehouse in Brooklyn has 1,000 books and its warehouse in Queens has 2,000 . Each store orders 1,500 books. It costs \(\$ 1\) to ship each book from Brooklyn to Manhattan and \(\$ 2\) to ship each book from Queens to Manhattan. It costs \(\$ 5\) to ship each book from Brooklyn to Long Island and \(\$ 4\) to ship each book from Queens to Long Island. a. If Big Red has a transportation budget of \(\$ 9,000\) and is willing to spend all of it, how many books should Big Red ship from each warehouse to each store in order to fill all the orders? b. Is there a way of doing this for less money? HINT [See Example 4.]
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