Problem 35 - Precalculus Mathematics for Calculus

Question

Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$\left\\{\begin{array}{l} x+4 y-2 z=-3 \\ 2 x-y+5 z=12 \\ 8 x+5 y+11 z=30 \end{array}\right.$$

Step-by-Step Solution

Answer

The system is dependent with solutions \(x = 5 - 2t\), \(y = t - 2\), \(z = t\).

Step 1: Write down the augmented matrix

First, express the given system of equations as an augmented matrix, which consolidates the coefficients and constant terms. The matrix is:\[\begin{bmatrix}1 & 4 & -2 & | & -3 \2 & -1 & 5 & | & 12 \8 & 5 & 11 & | & 30 \\end{bmatrix}\]

Step 2: Use Row Operations to Simplify

Apply Gaussian elimination to simplify the matrix into row-echelon form. Begin with eliminating the first column below the first row.- Replace row 2 with \(R2 - 2 \times R1\): \[ R2 = (2, -1, 5 \,|\, 12) - 2(1, 4, -2 \,|\, -3) = (0, -9, 9 \,|\, 18) \]- Replace row 3 with \(R3 - 8 \times R1\): \[ R3 = (8, 5, 11 \,|\, 30) - 8(1, 4, -2 \,|\, -3) = (0, -27, 27 \,|\, 54) \]

Step 3: Further Simplify the Rows

Continue simplifying the matrix. To simplify row 3 relative to row 2:- Replace row 3 with \(R3 - 3 \times R2\): \[ R3 = (0, -27, 27 \,|\, 54) - 3(0, -9, 9 \,|\, 18) = (0, 0, 0 \,|\, 0) \]

Step 4: Analyze the Row Conditions

After simplification, the matrix is:\[\begin{bmatrix}1 & 4 & -2 & | & -3 \0 & -9 & 9 & | & 18 \0 & 0 & 0 & | & 0 \\end{bmatrix}\]This result indicates a consistent system with infinitely many solutions because the last row is composed entirely of zeros.

Step 5: Express General Solution

Back-substitute and express the free variable (in this case, \(z\)) in terms of \(x\) and \(y\):- From row 2: \(-9y + 9z = 18\) simplifies to \(y - z = -2\) or \(y = z - 2\).- From row 1: \(x + 4y - 2z = -3\) becomes \(x = -3 - 4y + 2z\).Substituting for \(y\), \(x = -3 - 4(z - 2) + 2z = -3 - 4z + 8 + 2z = 5 - 2z\).Thus, the general solution is: \[\begin{align*}x &= 5 - 2t \y &= t - 2 \z &= t\end{align*}\]where \(t\) is any real number.

Key Concepts

Gaussian Elimination

Gaussian elimination is a mathematical procedure that simplifies a system of linear equations to make it easier to solve. This process transforms the system into a row-echelon form using row operations.
By doing this, you make it easy to see relationships between the variables. Here are some basic operations involved:

• Swapping one row with another
• Multiplying a row by a non-zero scalar
• Adding or subtracting the multiples of one row to another

In our exercise, Gaussian elimination helped us transform the original system into a simpler form. This step saves time and reduces error when solving for variables in complex systems.
After applying these transformations, you often find the system in a form ready for back-substitution, which then helps us determine the values of the variables explicitly. Understanding this process is fundamental because it's a key strategy for solving systems with multiple variables.

Augmented Matrix

An augmented matrix is a very convenient way to represent a system of linear equations. It combines the coefficients of the variables and the constants from each equation.
For instance, a system of equations is laid out in a tabular form, with the coefficients on the left and the constants after a vertical bar. This form lets you apply operations easily.
In our given problem, the augmented matrix was essential as it helped us perform Gaussian elimination directly on the entire system. This matrix is structured as follows:

• The first few columns show the coefficients of the variables.
• The final column, after the vertical line, shows the constant terms.

Using an augmented matrix, you can visually track the changes made during row operations. It's an integral part of simplifying systems efficiently to reach a solution quickly.

Inconsistent System

An inconsistent system of equations is one that has no solution. This happens when the lines represented by the equations do not intersect, meaning that they are parallel.
In terms of the augmented matrix, inconsistency is often revealed when row operations lead to an impossible equation, like 0 = 1. In our exercise, we did not encounter an inconsistent system.
Instead, the simplification led to a situation where a row became entirely zeros, indicating other characteristics. It’s crucial to recognize an inconsistent system as it will save time knowing no common solution exists. However, if a solution is needed, such a scenario often suggests re-evaluating the model for possible errors in the equations or relations.

Dependent System

A dependent system is characterized by having an infinite number of solutions. Here, the equations are linearly dependent, often meaning one equation is a combination of the others.
In our case, simplifying using Gaussian elimination led to a third row full of zeros, noting it's consistent yet dependent.
The solution implied another important aspect: free variables. These variables can take any value, leading to infinitely many solutions for the system of equations.

• The leading variables can be expressed in terms of free variables.
• The free variable acts like a parameter, often denoted by letters like \(t\).

This characteristic is very useful in real-world situations where a system doesn't just have one solution, but rather a range of possible solutions due to overlap or redundancy in the equations' formulations.