Problem 42 - Algebra for College Students

Question

In Problems 35-42, each matrix is the reduced echelon matrix for a system with variables \(x_{1}, x_{2}, x_{3}\), and \(x_{4}\). Find the solution set of each system. \(\left[\begin{array}{rrrr:r}1 & 0 & 0 & 0 & 7 \\ 0 & 1 & 0 & 0 & -3 \\ 0 & 0 & 1 & -2 & 5 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]\)

Step-by-Step Solution

Answer

\(x_1 = 7\), \(x_2 = -3\), \(x_3 = 5 + 2x_4\); \(x_4\) is free (any real number).

Step 1: Understand the Matrix Structure

The given matrix is in reduced echelon form, which provides information about the solution to the system of equations corresponding to the variables \(x_1, x_2, x_3,\) and \(x_4\). Each row represents an equation.

Step 2: Translate Rows into Equations

The first row translates to the equation \(x_1 = 7\), the second row translates to \(x_2 = -3\), and the third row translates to \(x_3 - 2x_4 = 5\). The fourth row is all zeros and doesn't contribute to any equation.

Step 3: Solve for Leading Variables

From the first two rows, we directly obtain \(x_1 = 7\) and \(x_2 = -3\). These are the specific solutions for those variables.

Step 4: Parameterize the Free Variable

The third row \(x_3 - 2x_4 = 5\) can be rewritten as \(x_3 = 5 + 2x_4\). Here, \(x_4\) can be any real number, making it a free variable in the solution set.

Step 5: Express the Solution Set

The solution set consists of specific values for \(x_1\) and \(x_2\), and an expression for \(x_3\) in terms of \(x_4\). The solution is: \( (x_1, x_2, x_3, x_4) = (7, -3, 5 + 2x_4, x_4) \) where \(x_4\) is any real number.

Key Concepts

Reduced Echelon Form

When solving systems of equations, one common representation is the matrix. A special form of this matrix is the reduced echelon form. This form is particularly handy because it makes finding solutions much easier. The reduced echelon form of a matrix has several distinctive features:

• The leading entry in each non-zero row is 1.
• Each leading 1 is the only non-zero entry in its column.
• The rows consisting entirely of zeros are at the bottom.
• The leading 1 of a non-zero row is always to the right of the leading 1 of the previous row.

Understanding these properties allows us to simplify solving the system of equations, as they visually separate the leading and free variables.

System of Equations

The core idea behind matrices in the context of linear algebra is to represent systems of equations. Each row of a matrix can be thought of as an equation in a system, with each column corresponding to a variable in those equations. For our example:

• The first row suggests the equation: \(x_1 = 7\).
• The second row corresponds to: \(x_2 = -3\).
• The third row gives us: \(x_3 - 2x_4 = 5\).
• The fourth row, being entirely zeroes, doesn't contribute new information.

By converting rows into equations, we translate matrix data into a more intuitive form for solving those equations.

Free Variable

In the context of a system of equations, a free variable is one that does not have a leading 1 in any row of the reduced echelon form matrix. This means it can take on any value.For our matrix, the variable \(x_4\) is a free variable because it does not have a unique equation that defines it. Instead, it is used to express other variables, such as in the equation \(x_3 = 5 + 2x_4\). By selecting different values for \(x_4\), we can determine a range of solutions for \(x_3\), which highlights the flexibility provided by free variables.

Solution Set

The solution set is the collection of all possible solutions to a system of equations. It describes all the ways the variables can be values that satisfy all equations in the system simultaneously.In our example, we have determined:

• \(x_1 = 7\), a fixed solution.
• \(x_2 = -3\), also fixed.
• \(x_3 = 5 + 2x_4\), which varies based on \(x_4\).
• \(x_4\) is not fixed, but a free variable that can be any real number.

Therefore, the solution set is represented as \((x_1, x_2, x_3, x_4) = (7, -3, 5 + 2x_4, x_4)\) where \(x_4\) can be any real number. This illustrates all potential solutions in a concise way, capturing the interrelationships between variables.