Problem 78 - Precalculus with Limits

Question

In Exercises 63-84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. \( \left\\{ \begin{array}{l} x + 2y = 0 \\ 2x + 4y = 0 \end{array} \right. \)

Step-by-Step Solution

Answer

The solution to the system of equations is all pairs \((-2y, y)\), where \(y\) is any real number.

Step 1: Form the Augmented Matrix

Form the augmented matrix from the system of equations. Each row corresponds to one equation and each column corresponds to a variable. The rightmost column represents the constants on the right side of the equations. Hence, for the given system of equations, the augmented matrix is \(\left[ \begin{array}{cc|c} 1 & 2 & 0 \ 2 & 4 & 0 \end{array} \right]\)

Step 2: Perform Gaussian Elimination or Gauss-Jordan Elimination

In this case, it can be observed that the second row is simply the first row multiplied by 2. Hence, subtracting 2 times the first row from the second row doesn't change the system. The resulting matrix is \(\left[ \begin{array}{cc|c} 1 & 2 & 0 \ 0 & 0 & 0 \end{array} \right]\)

Step 3: Interpret the Result

This resulting matrix corresponds to the system of equations \(x + 2y = 0\) and \(0 = 0\). The second equation is always true. It tells us that there are infinitely many solutions since it doesn't provide additional information about the variables. However, from the first equation, we can express \(x\) in terms of \(y\) as \(x = -2y\). The solution set of the system is then all pairs \((-2y, y)\), where \(y\) is any real number.

Key Concepts

Gauss-Jordan Elimination

Gauss-Jordan elimination is a method used to solve systems of linear equations by transforming the system's augmented matrix into a form called the reduced row echelon form (RREF). This process involves a series of row operations that simplify the matrix while preserving the original system's solutions. The three types of row operations used in this method are:

• Swapping two rows
• Multiplying a row by a nonzero scalar
• Adding or subtracting a multiple of one row to another row

In the example provided, the system of equations is transformed into an augmented matrix and simplified via these operations. When solving a system using Gauss-Jordan elimination, the goal is to make the leading coefficient (the first nonzero number from the left, in every row) equal to 1, and ensure that all other elements in that column are zero. This streamlined approach enables a direct reading of the solutions from the final matrix form.

Augmented Matrix

An augmented matrix is a crucial tool in solving systems of equations with computational methods. It efficiently organizes the variables and constants of the system in a compact matrix form. For a system like \(\begin{array}{l}x + 2y = 0 \2x + 4y = 0\end{array}\), the augmented matrix would be:\[ \begin{bmatrix} 1 & 2 & | & 0 \2 & 4 & | & 0 \end{bmatrix} \]Each row of this matrix corresponds to one equation, with columns representing the coefficients of variables followed by a column for the constant terms on the right side of the equations. The vertical bar separates the coefficients of the variables from the constants, indicating the '=' in the original equations. This setup allows for systematic manipulation using methods like Gauss-Jordan elimination to find solutions.

System of Equations

A system of equations is a set of two or more equations involving the same set of variables. Solving such a system means finding the values of the variables that satisfy all the equations simultaneously. In the provided example, the system is:

• \( x + 2y = 0 \)
• \( 2x + 4y = 0 \)

The goal is to determine the values for \(x\) and \(y\) that make both equations true. This particular system is linearly dependent, meaning both equations essentially state the same relationship between \(x\) and \(y\). Recognizing dependent equations can lead to solutions like infinite sets, or sometimes, no solutions, based on the setup.

Infinite Solutions

When a system of equations leads to infinitely many solutions, it typically means that the equations describe the same plane or line in the context of linear equations. This degeneracy happens when one equation can be transformed into another through multiplication or addition, indicating there isn't enough information to determine unique solutions for each variable.In the original problem:

• The system reduced to \( x + 2y = 0 \) and \( 0 = 0 \).

This system has infinite solutions because the second equation is always true and adds no new information, confirming that only the first equation is essential for solutions. The solutions can be expressed parametrically, such as \((x, y) = (-2y, y)\), where \(y\) can take any real number value, demonstrating the freedom in choosing \(y\). This reveals a line of solutions in the plane defined by these variables.