Problem 29 - College Algebra

Question

In Exercises \(29-32,\) 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. $$\left\\{\begin{array}{l}6 x+5 y=13 \\\5 x+4 y=10\end{array}\right.$$

Step-by-Step Solution

Answer

The system of equations can be written as a matrix equation as follows: \(\[\begin{array}{cc}6 & 5 \\5 & 4\end{array}\]\[\begin{array}{c}x \\y\end{array}\] = \[\begin{array}{c}13 \\10\end{array}\]\)

Step 1: Identify the Coefficients and Constants

The given system of equations is \(\{\begin{array}{l}6x+5y=13 \\5x+4y=10\end{array}\}\). Identify the coefficients of the variables, which in this case are 6, 5, 5 and 4. For each equation also identify the constants, which are 13 and 10.

Step 2: Form the Coefficient Matrix \(A\)

Build the matrix \(A\) by placing the coefficients of the variables \(x\) and \(y\) in it. The first row is formed by the coefficients of equation 1 and the second row is formed by the coefficients of equation 2. Hence, matrix \(A\) will be \(\[\begin{array}{cc}6 & 5 \\5 & 4\end{array}\]\)

Step 3: Form the Column Matrix \(B\)

The column matrix \(B\) is constructed by placing the constants 13 and 10 from the right side of the equations. Hence, matrix \(B\) will be \(\[\begin{array}{c}13 \\10\end{array}\]\)

Step 4: Write the Matrix Equation

Rewrite the system in terms of a matrix equation of the form \(AX = B\). The unknown vector \(X\) will contain the variables \(x\) and \(y\). Substituting in the matrices \(A\) and \(B\) and unknown vector \(X\) formed, the matrix equation is \(\[\begin{array}{cc}6 & 5 \\5 & 4\end{array}\]\[\begin{array}{c}x \\y\end{array}\] = \[\begin{array}{c}13 \\10\end{array}\]\)

Key Concepts

Linear Algebra

Linear algebra is a branch of mathematics concerned with vectors, vector spaces (also known as linear spaces), linear transformations, and systems of linear equations.

It involves the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.

A key operation in linear algebra is the multiplication of a matrix by a vector, which results in a new vector. Linear algebra is not just about solving linear equations, but understanding transformations and structures that extend across many mathematical areas.

Applications of linear algebra can be found in various fields including science, engineering, computer graphics, social sciences, and economics due to its ability to handle complex operations and multidimensional scenarios.

Coefficient Matrix

The coefficient matrix is a rectangular array of numbers that represents a system of linear equations. Each row corresponds to an equation, and each column corresponds to a variable.

In the system from our exercise, variables are multiplied by their respective coefficients and the collection of these coefficients forms the coefficient matrix, denoted as matrix A. As part of the matrix equation, it plays a crucial role in the solution of systems of equations using matrix operations.

These matrices are fundamental in transforming geometric data, encrypting communications, or even in algorithms for internet searches. Understanding and manipulating coefficient matrices enable solving multiple equations simultaneously, which is more efficient than tackling them individually.

System of Equations

A system of equations is a set of two or more equations that share a common set of variables. The goal is to find the values for the variables that satisfy all equations in the system simultaneously.

In the context of the exercise provided, the equations are linear, and the objective is to write the system in the form of a matrix equation. This transforms a problem that may seem complex into a more manageable format that can be solved using matrix operations.

Solving a system of equations can be interpreted as finding the point(s) at which the graphs of the equations intersect. These solutions can represent real-world quantities, such as the intersection of paths, the equilibrium of forces, or the balance of financial transactions.