Problem 7 - A Graphical Approach to Precalculus with Limits

Question

Write the augmented matrix for each system. Do not solve the system. $$\begin{array}{r} 2 x+3 y=11 \\ x+2 y=8 \end{array}$$

Step-by-Step Solution

Answer

The augmented matrix is \(\begin{bmatrix} 2 & 3 & | & 11 \\ 1 & 2 & | & 8 \end{bmatrix}\).

Step 1: Understand the System of Equations

First, we have a system of linear equations consisting of two equations. The system is: \(2x + 3y = 11\) and \(x + 2y = 8\). Each equation represents a line in a 2-dimensional space.

Step 2: Identify Coefficients and Constants

Next, we identify the coefficients of the variables and the constants from each equation. From equation 1: \(2x + 3y = 11\), the coefficients are \(2\) for \(x\), \(3\) for \(y\), and the constant is \(11\). For equation 2: \(x + 2y = 8\), the coefficients are \(1\) for \(x\), \(2\) for \(y\), and the constant is \(8\).

Step 3: Construct the Augmented Matrix

An augmented matrix is a compact way to represent a system of linear equations. It consists of the coefficients of the variables and the constants from the equations systematically arranged. For the given system, the augmented matrix is constructed as \[ \begin{bmatrix} 2 & 3 & | & 11 \ 1 & 2 & | & 8 \end{bmatrix} \]. Here, each row corresponds to an equation, and the vertical line separates the coefficients and the constants.

Key Concepts

Understanding the System of Equations

A system of equations is a set of two or more equations that share two or more variables. These equations are considered simultaneously because they describe relationships where the same variables are involved. For instance, consider the system:

• \(2x + 3y = 11\)
• \(x + 2y = 8\)

Each equation in this system represents a line on a two-dimensional graph, and the solution to this system is the point where these lines intersect. The main goal in working with a system of equations is to find values for the variables that satisfy all the equations at the same time.
Systems can be solved using various methods, such as substitution, elimination, or by using matrices. In our context, we focus on representing the system with an augmented matrix, which serves as an efficient way to handle and manipulate these equations.

Exploring Linear Equations

Linear equations are equations where the highest power of the variable is one. They form straight lines when graphed on the coordinate plane. The general form of a linear equation in two variables is \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables.

In our given system:

• The first equation is \(2x + 3y = 11\), with "2x" and "3y" forming a linear combination resulting in the constant 11.
• The second equation is \(x + 2y = 8\), following the same linear pattern with a different set of coefficients and constant.

Linear equations allow us to model relationships where one quantity depends linearly on another, and because of their simplicity, they are fundamental in both algebra and real-world applications.

Identifying Coefficients and Constants

In the context of linear equations and systems, coefficients and constants play crucial roles. The **coefficients** are the numbers that multiply the variables (like \(x\) and \(y\)), while the **constants** are the standalone numbers. Let's break it down with our example.

• For the equation \(2x + 3y = 11\):
  • The coefficient of \(x\) is 2.
  • The coefficient of \(y\) is 3.
  • The constant is 11.
• For the equation \(x + 2y = 8\):
  • The coefficient of \(x\) is 1 (often implied).
  • The coefficient of \(y\) is 2.
  • The constant is 8.

Recognizing these elements helps in the formulation of the augmented matrix, where they are laid out in rows to simplify further operations. Understanding coefficients and constants is essential as they determine the slope and position of the line represented by each linear equation.