Problem 40 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Write the given system of linear equations in matrix form. \(\begin{aligned} 3 x_{1}-5 x_{2}+4 x_{3}=& 10 \\ 4 x_{1}+2 x_{2}-3 x_{3}=&-12 \\\\-x_{1} \quad+x_{3}=&-2 \end{aligned}\)

Step-by-Step Solution

Answer

The given system of linear equations can be written in matrix form as: \[\begin{pmatrix} 3 & -5 & 4 \\ 4 & 2 & -3 \\ -1 & 0 & 1 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 10 \\ -12 \\ -2 \end{pmatrix}\]

Step 1: Identify the Coefficient Matrix

First, we'll identify the coefficients of the variables in the given system of linear equations and organize them in a 3x3 matrix \(A\). The matrix should look like this: \[A = \begin{pmatrix} 3 & -5 & 4 \\ 4 & 2 & -3 \\ -1 & 0 & 1 \end{pmatrix}\]

Step 2: Identify the Variables Matrix

Now, we'll identify the variables matrix \(x\), which is a 3x1 matrix containing the variables \(x_1, x_2, x_3\). The matrix should look like this: \[x = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}\]

Step 3: Identify the Solution Matrix

Finally, we'll identify the solution matrix \(b\), which is a 3x1 matrix containing the constants on the right-hand side of the given system of linear equations. The matrix should look like this: \[b = \begin{pmatrix} 10 \\ -12 \\ -2 \end{pmatrix}\]

Step 4: Write the Matrix Form of the System of Equations

We now have all the necessary matrices to write the system of linear equations in matrix form, using the general format \(Ax = b\). The final result is: \[\begin{pmatrix} 3 & -5 & 4 \\ 4 & 2 & -3 \\ -1 & 0 & 1 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 10 \\ -12 \\ -2 \end{pmatrix}\]

Key Concepts

Matrix Form Representation

Visualizing a system of linear equations in matrix form simplifies the process of finding solutions. It involves writing the equations in a compact manner that aligns coefficients and variables in structures known as matrices. This method leverages matrix operations to solve for unknowns much like one would solve algebraic equations.

The key benefit of matrix form is that it paves the way for using computational tools to solve systems that might otherwise be too complex to solve by hand. For example, the equation set from our exercise \[\begin{aligned} 3 x_{1}-5 x_{2}+4 x_{3}=& 10 \ 4 x_{1}+2 x_{2}-3 x_{3}=&-12 \-x_{1} + x_{3}=&-2 \end{aligned}\] can be represented as \[Ax = b\], which is a concise and powerful expression that carries all the information of the system.

Coefficient Matrix

The coefficient matrix, often denoted as \(A\), captures all the multipliers of the variables in the system. It is a grid where each row represents the coefficients from a single equation and each column corresponds to a specific variable across all equations.

In our exercise, the coefficient matrix is a 3x3 matrix because there are three equations and three variables. It looks like this: \[A = \begin{pmatrix}3 & -5 & 4 \4 & 2 & -3 \-1 & 0 & 1\end{pmatrix}\].
Understanding the structure of the coefficient matrix is crucial as it directly affects the methods and ease of solving the system.

Variables Matrix

The variables matrix, denoted by \(x\), is a column matrix that lists all of the variables of the system in order. For our given system of equations, this matrix is made up of the variables \(x_1\), \(x_2\), and \(x_3\), as follows: \[x = \begin{pmatrix}x_1 \x_2 \x_3\end{pmatrix}\].

This matrix is pivotal because it represents the unknowns we're trying to solve for. In practical terms, finding the solution to the system means calculating the correct values for each entry in this variables matrix.

Solution Matrix

The solution matrix, typically denoted by \(b\), collects the constants from the right side of each equation in the system. It reflects the outcomes that the linear combinations of the variables aim to achieve.

In the example problem, the solution matrix is: \[b = \begin{pmatrix}10 \-12 \-2\end{pmatrix}\].
The arrangement into a matrix format allows us to move towards finding the values for \(x_1\), \(x_2\), and \(x_3\) that satisfy all equations simultaneously, which is what solving a system of linear equations is all about.