Problem 70 - 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. \( \left\\{ \begin{array}{l} x - 3y = 5 \\ -2x + 6y = -10 \end{array} \right. \)

Step-by-Step Solution

Answer

The system of equations has infinitely many solutions.

Step 1: Set up the augmented matrix

We can represent this system of equations with a 2x3 matrix, or an augmented matrix. The left side of the matrix represents the coefficients of the variables, and the right side represents the constants. Our matrix for the system \[ \begin{pmatrix} 1 & -3 & 5 \\ -2 & 6 & -10 \end{pmatrix} \]

Step 2: Perform row operations

The matrix represents the system of equations, so we can perform operations on the rows of the matrix (which correspond to the equations) to simplify it. To eliminate \(x\) from the second equation, we can add twice the first row to the second row: \[ \begin{pmatrix} 1 & -3 & 5 \\ 0 & 0 & 0 \end{pmatrix} \]

Step 3: Interpret the resulting system of equations

The second row of our matrix is now all zeros, which corresponds to a equation of 0=0, this is always true. This means that there are many solutions to this system of equations, depending on the value of \(y\) chosen, \(x\) can be found from the first equation. So, the system has infinitely many solutions.

Key Concepts

Gauss-Jordan Elimination

Gauss-Jordan Elimination is a powerful method for solving systems of linear equations. This technique systematically transforms a given augmented matrix into reduced row-echelon form using a series of row operations. The goal is to make the left side of the augmented matrix, which consists of the coefficients of the variables, into an identity matrix.
Once the left side becomes an identity matrix, the solutions to the system of equations can directly be read from the right side of the matrix. This method offers the advantage of finding solutions without the need for back-substitution, making it a preferred technique in many scenarios.

• Start with an augmented matrix representing the system of equations.
• Perform a sequence of operations on rows: swap rows, multiply rows by non-zero constants, or add multiples of rows to each other.
• Aim to construct an identity matrix on the left side of the augmented matrix.
• Once the left is the identity matrix, the right contains the solutions.

Even though Gauss-Jordan Elimination involves straightforward steps, it often requires precision and attentiveness to avoid errors during row operations.

Augmented Matrix

An augmented matrix is a crucial component in solving systems of linear equations. It is essentially a compact representation where the coefficients and constants of the equations are combined into a single matrix. This matrix format facilitates the application of row operations, streamlining the process of finding solutions.
To create an augmented matrix from a system of equations:

• Align variables in each equation consistently into columns.
• Write down the coefficients in the left part of the matrix.
• Place the constants in the rightmost column of the matrix.

For example, the system:- _x - 3y = 5_- _-2x + 6y = -10_can be converted to the augmented matrix:\[\begin{pmatrix}1 & -3 & 5 \-2 & 6 & -10\end{pmatrix}\]This representation is both concise and ideal for performing the various operations needed in methods like Gaussian elimination and Gauss-Jordan elimination.

System of Equations

A system of equations is a collection of two or more equations with a shared set of variables. The primary aim when working with these systems is finding values for the variables that simultaneously satisfy all the given equations.
Systems can be categorized into different types based on their solutions:

• Consistent: At least one solution exists, which may be a single solution or infinitely many.
• Inconsistent: No solutions exist; the equations contradict each other.

In the original exercise provided, the system:- _x - 3y = 5_- _-2x + 6y = -10_results in an augmented matrix that leads to a special scenario:\[\begin{pmatrix}1 & -3 & 5 \0 & 0 & 0\end{pmatrix}\]This indicates the system is consistent with infinitely many solutions, as the second row translates to a true statement (0=0). As a result, the value of \( y \) can be freely chosen, and \( x \) will adjust accordingly based on the first equation.