Problem 32 - Algebra and Trigonometry

Question

Write each linear system as a matrix equation in the form \(A X=B\), where \(A\) is the coefficient matrix and \(B\) is the constant matrix. $$ \text { 32. }\left\\{\begin{aligned} x+4 y-z &=3 \\ x+3 y-2 z &=5 \\ 2 x+7 y-5 z &=12 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The matrix form of the given system of equations is given by \(A X = B\), where \(A = \[[1, 4, -1], [1, 3, -2], [2, 7, -5]\]\), \(X = \[[x], [y], [z]\]\), and \(B = \[[3], [5], [12]\]\).

Step 1: Identify matrix A

The coefficient matrix \(A\) is consisting of the coefficients of the variables \(x\), \(y\), and \(z\). In our given system, we can read off these coefficients to create the matrix A, which becomes \(A = \[[1, 4, -1], [1, 3, -2], [2, 7, -5]\]\).

Step 2: Identify matrix X

The solution matrix \(X\) consists of the variables of the system. In this case, the variables are \(x\), \(y\), and \(z\), so our matrix \(X\) becomes \(X = \[[x], [y], [z]\]\).

Step 3: Identify matrix B

The constant matrix \(B\) consists of the constants on the right side of the system equations. These can be read off from the system to form the matrix \(B\), which becomes \(B = \[[3], [5], [12]\]\).

Key Concepts

Linear Systems

Linear systems are sets of equations where each equation is linear. Linear means that the variables are only to the first power and combined using operators like addition and subtraction. For example, in the equations provided in the problem, we see each equation fits these criteria. Linear systems can have infinite solutions, one solution, or no solution. To solve them, we often use methods like substitution, elimination, or matrix equations.

• They typically appear in the form of several equations together.
• They involve relationships represented by straight lines when graphed.

Understanding linear systems is crucial because they form the basis for more complex algebraic concepts, and provide a foundational understanding of relationships and equations that appear in higher-level mathematics and various applications, such as physics and engineering.

Coefficient Matrix

A coefficient matrix is a matrix consisting solely of the coefficients of the variables from each equation in a linear system. In the context of matrices, coefficients are the numbers that multiply the variables.
In our example, the coefficient matrix, denoted as \( A \), is comprised of the numerical coefficients for \( x \), \( y \), and \( z \) from each equation of the linear system.

• It describes how each variable contributes to each equation.
• It's a neat way to organize information, making it easier for calculations with computers or by hand.

This matrix is vital when converting a system of equations into matrix form, simplifying the process of using matrix operations to find solutions.

Constant Matrix

The constant matrix, denoted here as \( B \), consists of the constants from the equations in the linear system. These constants are the numbers without variables on the right side of each equation. Simply put, they are what you get when substituting all variables on one side of the equation.
In our given system:

• The constants are 3, 5, and 12.
• These form a single column matrix, aligning with each equation.

The role of the constant matrix is crucial because it represents the equality we are trying to balance in the equations. When writing out a matrix equation, the goal is to find values for the variables that satisfy this equality. The constant matrixhelps in framing the problem as a matrix equation \( A X = B \). This form can then be tackled through various matrix-solving methods to find the solution.