Problem 46 - Precalculus: Mathematics for Calculus

Question

Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$\left\\{\begin{aligned} 3 r+2 s-3 t &=10 \\ r-s-t &=-5 \\ r+4 s-t &=20 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system is dependent with the solution \( r = 2t - 5, s = t, t \) real number.

Step 1: Write the System in Matrix Form

First, represent the given system of linear equations as an augmented matrix. The system is: \[ \begin{align*} 3r + 2s - 3t &= 10 \ r - s - t &= -5 \ r + 4s - t &= 20 \end{align*} \] So, the augmented matrix is: \[ \begin{bmatrix} 3 & 2 & -3 & | & 10 \ 1 & -1 & -1 & | & -5 \ 1 & 4 & -1 & | & 20 \end{bmatrix} \]

Step 2: Use Row Operations to Simplify

Perform row operations to transform the matrix into a simpler form. First, we can subtract Row 2 from Row 1 by multiplying Row 2 by 3 (for better clarity): Row 1 becomes \(R1 - 3R2\), which leads us to: \[ \begin{bmatrix} 0 & 5 & 0 & | & 25 \ 1 & -1 & -1 & | & -5 \ 1 & 4 & -1 & | & 20 \end{bmatrix} \]. Now subtract Row 2 from Row 3 to obtain: \(R3 - R2\) giving us \[ \begin{bmatrix} 0 & 5 & 0 & | & 25 \ 1 & -1 & -1 & | & -5 \ 0 & 5 & 0 & | & 25 \end{bmatrix} \].

Step 3: Identify Consistency or Dependency

Looking at the modified matrix, we see that Rows 1 and 3 are identical. Since the two rows are the same, this shows dependency between the equations. Specifically, the row \(0x + 5y + 0z = 25\) repeats, indicating infinitely many solutions rather than a contradiction or inconsistency.

Step 4: Express the Solution in Parametric Form

The system is dependent, so we parameterize the solution. Let's express any variable in terms of a parameter, for simplicity, define \(s = t\), then we solve for \(r\) and \(s\) in terms of \(t\). From Row 2: \(r = s + t - 5\) and since \(s = t\), substitution gives \(r = t + t - 5 = 2t - 5\). Hence, solutions can be expressed as \(r = 2t - 5\), \(s = t\), and \(t\) is any real number.

Key Concepts

Augmented Matrix

An augmented matrix is a compact way to represent a system of linear equations. It combines the coefficients of variables and the constants from the equations into a single matrix. This makes it easier to perform calculations and transform the system using matrix operations.

In the problem given, the system of three linear equations is represented as:

• Equation 1: \(3r + 2s - 3t = 10\)
• Equation 2: \(r - s - t = -5\)
• Equation 3: \(r + 4s - t = 20\)

By arranging these equations into matrix form, you get:\[\begin{bmatrix}3 & 2 & -3 & | & 10 \1 & -1 & -1 & | & -5 \1 & 4 & -1 & | & 20\end{bmatrix}\]Here, the vertical line (|) separates the coefficients from the constants. This augmented matrix provides a structured way to proceed with further manipulation steps.

Row Operations

Row operations are transformations that you can perform on the rows of a matrix to simplify or solve a system of equations. There are three main types of row operations, and they are very useful:

• Swap the positions of two rows.
• Multiply all terms of a row by a non-zero constant.
• Add or subtract the multiple of one row from another row.

These operations do not alter the solution of the system, ensuring that the integrity of the original equations is preserved.

In our specific example, we used row operations to simplify the original augmented matrix to a form where we could easily identify dependencies between the equations. For instance, subtracting a multiple of one row from another can help remove variables, making it easier to spot relationships or solve for particular variables.

Dependent System

A system of linear equations is called dependent if at least one of the equations can be derived from others using basic algebraic operations (like addition or subtraction). This implies that the equations are not independent and share common solutions.

In the task at hand, when we applied row operations, we noticed that two rows in the augmented matrix became identical. This indicated that one of the equations was essentially a repetition of another.

Such dependency signifies that the system does not have a single unique solution. Instead, there are infinitely many solutions since the equations describe the same plane or line in space. Identifying dependency is crucial because it sets the basis for finding the complete parametric solution for the system.

Infinite Solutions

In a dependent system like the one given, infinite solutions occur because there is more than one way to satisfy all equations with the same values of variables.

Once dependency was identified, the next step was to express the system's solutions in a parametric form. This involves selecting one of the variables as a parameter—often choosing the variable that simplifies expressions further.

In this example, setting \(s = t\) allows us to express the remaining variables in terms of \(t\):

• \(r = 2t - 5\)
• \(s = t\)
• \(t\) is any real number.

This solution set encapsulates infinitely many solutions, as \(t\) can vary over all real numbers, yielding different combinations of \(r, s,\) and \(t\) that satisfy all the original equations simultaneously.