Question

Describe how you would explain to another student that the augmented matrix below represents a dependent system of equations. Describe a way to write the infinitely many solutions of this system. \(\left[\begin{array}{rrrrr}1 & -2 & 3 & -6 \\ 0 & 1 & 2 & \vdots & 5 \\ 0 & 0 & 0 & 0\end{array}\right]\)

Short answer

The given matrix represents a dependent system of equations due to the row of zeros at the bottom. This means that we have an equation that is not providing any new information. Infinitely many solutions to this system can be represented by at least one variable being free to take any value due to less equations than variables. Expressing the other variables in terms of this free variable gives the general solution.

Step-by-step solution

Recognize the Dependent System

Looking at the given matrix, the first thing to note is the row of zeros in the bottom of the augmented matrix, which indicate there being infinitely many solutions to the system of equations. This is because these zeros represent a redundant equation, i.e., an equation that doesn't provide any new information about our variables. This means our system is a dependent system.

Describe the Infinitely Many Solutions

Move to showing how to describe the infinitely many solution. Recall that for systems with infinitely many solutions, at least one variable is free, meaning it can take any value. The system has less equations than variables, thus allowing the free variable. In the given system, the last variable is free as it doesn't have an equation in the matrix. Therefore, consider it \(s\). Expressing the other variables in terms of \(s\) gives the general solution.

Formulating the General Solution

Let the free variable be \(s = t\) (where t can be any real number), now we plug \(s = t\) back into the second row equation which gives us the value for second variable \(r = 5 - 2t\). Finally, plugging the values of \(s\) and \(r\) back into the first row equation we find that the first variable \(p = 6 + 2t\). Therefore, the general solution is expressed in parametric form as \(p = 6+2t\), \(r = 5-2t\), and \(s = t\) where \(t\) is any real number.

Key concepts

Augmented Matrix

An **augmented matrix** is a convenient way to organize a system of linear equations. It consists of the coefficients of variables and the constants from each equation, represented as rows. In this matrix, the separation between the variables' coefficients and constants is marked by a vertical line. This setup helps visualize and solve the system systematically.

Consider the matrix provided in the exercise:

• Row 1: Represents the equation: \( x_1 - 2x_2 + 3x_3 = -6 \)
• Row 2: Represents the equation: \( x_2 + 2x_3 = 5 \)
• Row 3: Is all zeros, indicating that it adds no new information.

The presence of the row of zeros confirms we have a dependent system. This means one of the equations is redundant, leading the system to have infinitely many solutions.

Infinitely Many Solutions

In a **dependent system of equations**, not all equations add new information. As a result, the solution isn't unique. The system is consistent, but because of redundancy, there are infinitely many solutions.

From an augmented matrix perspective:

• A row of zeros suggests at least one variable can take any value, leading to multiple solutions.

To explain this effectively: - Since there's a row of zeros, not all conditions restrict the variables. - The augmented matrix system has fewer independent rows than variables, causing dependency.

Therefore, when we solve the system, it will have one or more free variables instead of fixed values for all variables. Thus, allowing the system to have infinitely many possible solutions.

Parametric Form

The solutions to a dependent system can often be expressed in **parametric form**. This involves setting one or more variables as parameters, typically represented by a variable such as \( t \), which can take any real value. Then, express the other variables in terms of these parameters.

For our system, assuming \( s = t \) where \( s \) is the free variable:

• The second row equation \( x_2 + 2x_3 = 5 \) becomes \( r = 5 - 2t \).
• Similarly, the first row equation gives us \( p = 6 + 2t \).

Thus, the solution set can be written as: - \( p = 6 + 2t \)- \( r = 5 - 2t \)- \( s = t \)

This form indicates that for any real value of \( t \), we have a corresponding set of solutions \((p, r, s)\) satisfying all the original equations.