Problem 59 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solving a System of Linear Equations (a) write the system of equations as a matrix equation \(A X=B\) and (b) use Gauss-Jordan elimination on the augmented matrix \([A: B]\) to solve for the matrix X. Use a graphing utility to check your solution. $$\left\\{\begin{aligned} -x_{1}+x_{2} &=4 \\ -2 x_{1}+x_{2} &=0 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution to the given system of linear equations is \(x_1 = 0\) and \(x_2 = 8\).

Step 1: Writing the System of Equations as a Matrix Equation

The equations \( -x_1 + x_2 = 4 \) and \(-2x_1 + x_2 = 0 \) can be transformed into a matrix equation of the form A X = B as follows. Let matrix A be the coefficients of \(x_1\) and \(x_2\), matrix X consist of the unknowns \(x_1\) and \(x_2\) and matrix B hold the constants on the other side of the equation. Therefore, matrix A= \[ \begin{pmatrix} -1 & 1 \ -2 & 1 \end{pmatrix} \], Matrix X = \[ \begin{pmatrix} x_1 \ x_2 \end{pmatrix} \] And Matrix B = \[ \begin{pmatrix} 4 \ 0 \end{pmatrix} \] So we have the matrix equation, \( AX = B \)

Step 2: Applying Gauss-Jordan elimination

We form the augmented matrix [A : B], which reads as \[ \begin{pmatrix} -1 & 1 & | & 4 \ -2 & 1 & | & 0 \end{pmatrix} \] We can start the Gauss-Jordan elimination process by swapping the first and the second row to get rid of the negative sign in the first row: we get \[ \begin{pmatrix} -2 & 1 & | & 0 \ -1 & 1 & | & 4 \end{pmatrix} \] Then add the first row to the second row and multiply the first row by -1/2. We get \[ \begin{pmatrix} 1 & -1/2 & | & 0 \ 0 & 1/2 & | & 4 \end{pmatrix} \] Now multiply the second row by 2 for convenience. We get \[ \begin{pmatrix} 1 & -1/2 & | & 0 \ 0 & 1 & | & 8 \end{pmatrix} \] Finally, add the first row to the second row. This implies \[ \begin{pmatrix} 1 & 0 & | & 0 \ 0 & 1 & | & 8 \end{pmatrix} \] This implies \(x_1 = 0\) and \(x_2 = 8\) is the solution to the system of equations.

Step 3: Checking the Result

We can now confirm our results by substituting \(x_1 = 0\) and \(x_2 = 8\) into the original system of equations: \( -x_1 + x_2 = 4 \) and \(-2x_1 + x_2 = 0 \). If our solution is correct, it should satisfy both equations.

Key Concepts

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. Here, the aim is to find values for the variables that satisfy every equation in the system.
In our example exercise, we are dealing with the system:

• \( -x_1 + x_2 = 4 \)
• \( -2x_1 + x_2 = 0 \)

When solving such systems, we seek a common set of values for \(x_1\) and \(x_2\) that meet both equations simultaneously. These systems can often be visualized as the intersection points of lines in a plane.

Matrix Equation

A matrix equation is a compact way of expressing systems of linear equations. They are written in the form \(AX = B\), where \(A\) is a matrix containing coefficients, \(X\) is a column matrix of variables, and \(B\) is a column matrix of constants.
In the provided exercise, we transform our system of equations into:

• Matrix \(A = \begin{pmatrix} -1 & 1 \ -2 & 1 \end{pmatrix}\)
• Matrix \(X = \begin{pmatrix} x_1 \ x_2 \end{pmatrix}\)
• Matrix \(B = \begin{pmatrix} 4 \ 0 \end{pmatrix}\)

This method of using matrices helps facilitate calculations, especially with larger systems, and sets the stage for methods such as Gauss-Jordan elimination.

Augmented Matrix

An augmented matrix is an efficient way to represent a system of linear equations and their constants in one entity.
For the system in our exercise, the augmented matrix combines the coefficient matrix \(A\) and the constant matrix \(B\) into:

• \(\begin{pmatrix} -1 & 1 & | & 4 \ -2 & 1 & | & 0 \end{pmatrix}\)

The vertical line in the matrix is used to separate the coefficients from the constants.
This format is valuable because it prepares the system for methods like Gauss-Jordan elimination, which systematically simplifies equations to find solutions.

Graphing Utility

A graphing utility is a tool, often a software or calculator feature, that graphically represents equations and systems of equations. It serves as a visual confirmation of solutions found algebraically.
To use a graphing utility for the given system, input each equation into the utility:

• Plot \( -x_1 + x_2 = 4 \)
• Plot \( -2x_1 + x_2 = 0 \)

By graphing both lines, you can visually identify the point of intersection. This point represents the solution to the system. In our case, it should validate the solution \(x_1 = 0\) and \(x_2 = 8\), showing where both equations are satisfied simultaneously.