Problem 46 - College Algebra with Modeling and Visualization

Question

Represent the system of linear equations in the form \(A X=B\) \(\begin{aligned} x-2 y+z &=12 \\ 4 y+3 z &=13 \\\\-2 x+7 y &=-2 \end{aligned}\)

Step-by-Step Solution

Answer

Matrices are: \(A = \begin{bmatrix} 1 & -2 & 1 \\ 0 & 4 & 3 \\ -2 & 7 & 0 \end{bmatrix}, X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, B = \begin{bmatrix} 12 \\ 13 \\ -2 \end{bmatrix}\) and the system is \(AX=B\).

Step 1: Identify Matrix A

The matrix \(A\) represents the coefficients of the variables (\(x, y, z\)) in the system of equations. It can be constructed by placing each coefficient in its respective position, corresponding to each variable's row and column.From the equations, we get:- From the first equation, the coefficients are \(1, -2,\) and \(1\).- From the second equation, the coefficients are \(0, 4,\) and \(3\).- From the third equation, the coefficients are \(-2, 7,\) and \(0\).Thus, matrix \(A\) is:\[A = \begin{bmatrix} 1 & -2 & 1 \ 0 & 4 & 3 \ -2 & 7 & 0 \end{bmatrix}\]

Step 2: Identify Matrix X

Matrix \(X\) represents the variables of the equation system. In this case, the variables are \(x, y,\) and \(z\). Thus, matrix \(X\) is:\[X = \begin{bmatrix} x \ y \ z \end{bmatrix}\]

Step 3: Identify Matrix B

Matrix \(B\) represents the constants on the right side of each equation in the system. These constants are:- From the first equation, the constant is \(12\).- From the second equation, the constant is \(13\).- From the third equation, the constant is \(-2\).Thus, matrix \(B\) is:\[B = \begin{bmatrix} 12 \ 13 \ -2 \end{bmatrix}\]

Step 4: Write the System as AX = B

Combine the matrices \(A\), \(X\), and \(B\) to represent the system of equations in the form \(A X = B\).The system is:\[\begin{bmatrix} 1 & -2 & 1 \ 0 & 4 & 3 \ -2 & 7 & 0 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 12 \ 13 \ -2 \end{bmatrix}\]

Key Concepts

Matrix Representation

The concept of matrix representation is a fundamental idea in mathematics, especially in solving systems of linear equations. It allows us to organize and solve equations in a structured manner. A matrix is essentially a rectangular array of numbers or variables arranged in rows and columns.
This layout helps us to manipulate and solve systems efficiently by using operations like addition, subtraction, and multiplication.
In the context of a system of linear equations, matrices can represent both the coefficients of variables and the constants in equations. For example, in our system of equations:

• The matrix \(A\) is constructed using the coefficients of the variables \(x, y,\) and \(z\).
• The matrix \(X\) holds the variables themselves.
• Matrix \(B\) is comprised of the constants from each equation.

By arranging equations into this matrix form, we efficiently capture the entirety of the system within a mathematical structure, providing clarity and a path toward a solution.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, matrices, and linear transformations. One of its key uses is in solving systems of linear equations, just as we do with matrix representation.
Linear equations can be thought of as straight lines when graphed, and when we combine multiple such equations, they may intersect at a common solution point.
In linear algebra, the method of representing systems of equations as matrices (as demonstrated in our example) allows us to apply powerful techniques like Gaussian elimination or determinant calculus to find solutions. This structured approach is what makes linear algebra a versatile tool in sciences and engineering.
Through matrices like \(A, X,\) and \(B\), linear algebra helps to simplify the complexity of handling multiple linear equations, providing us with elegant solutions without dealing explicitly with each equation individually.

Variables and Coefficients

Variables and coefficients are core components within any system of equations. - **Variables** are the unknowns in the equations we are trying to solve. In our example, these variables are \(x, y,\) and \(z\). They represent potential values that can make all equations in the system true at the same time.
- **Coefficients** are the numerical factors that multiply the variables. They dictate the weight or influence each variable has within its equation. In the matrix \(A\), every entry represents a coefficient linked to a particular variable.Understanding how coefficients and variables interact is crucial for solving any system of equations. Recognizing that coefficients position the variables can allow them to crystallize into a solution and provide insights into the relationships and dependencies between variables across different equations.Through careful organization, as seen in matrix \(A\), the clarity provided can pave the path for finding answers, helping to solve even complex systems.