Problem 7 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

Augmented matrix: \( \begin{bmatrix} 2 & 3 & \vert & 11 \\ 1 & 2 & \vert & 8 \end{bmatrix} \)

Step 1: Identify Coefficients and Constants

First, identify the coefficients and constants from each equation. For the first equation, \(2x + 3y = 11\), the coefficients are 2 and 3, with 11 as the constant. For the second equation, \(x + 2y = 8\), the coefficients are 1 and 2, and the constant is 8.

Step 2: Formulate Augmented Matrix

Write the augmented matrix using the coefficients and constants from the identified system. The augmented matrix includes the coefficients of each variable and the constants as a separate column, resulting in the format: \[\begin{bmatrix} 2 & 3 & \vert & 11 \ 1 & 2 & \vert & 8 \end{bmatrix} \]

Key Concepts

Systems of Equations

A system of equations is a set of two or more equations having the same set of variables. They are a powerful mathematical tool for solving real-life problems where multiple conditions are given simultaneously. For example, in our exercise, we have two equations with variables \(x\) and \(y\):

• \(2x + 3y = 11\)
• \(x + 2y = 8\)

The goal when working with systems of equations is often to find values of the variables that satisfy all equations at the same time. However, in this exercise, we aren't solving the system, but representing it in a matrix form known as an "augmented matrix." By visualizing these relationships, it becomes easier to manipulate and solve the equations when necessary. Using techniques such as substitution, elimination, or matrix operations, we can determine the values of \(x\) and \(y\) that resolve the system.

Matrix Representation

Matrix representation is an organized way of presenting information, especially useful for systems of equations. In the context of the exercise, we convert the system of equations into a matrix format, creating what is called an augmented matrix. An augmented matrix includes coefficients of variables and constants from the equations grouped into a rectangular array.
For the equations:

• \(2x + 3y = 11\)
• \(x + 2y = 8\)

The augmented matrix would look like this:\[\begin{bmatrix}2 & 3 & \vert & 11 \1 & 2 & \vert & 8\end{bmatrix}\]This matrix consists of rows and columns where each row corresponds to an equation, and each column represents a coefficient or constant. The vertical line separates the coefficients from the constants. This visual format makes it easier to use matrix operations to solve the system, a method commonly used in linear algebra.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and operation symbols, used to represent real-world scenarios. In the context of our exercise, each individual equation like \(2x + 3y = 11\) is an algebraic expression that relates variables \(x\) and \(y\) with constants. They provide a concise way to convey mathematical relationships and equations.
In a system of equations:

• Each equation is treated as an algebraic expression.
• They can be transformed and manipulated to find the values of variables, which makes rooting out solutions systematic and structured.
• Variables in the expressions are symbolic placeholders that can represent unknown or varying values. Constants like \(11\) and \(8\) in our system are fixed numbers they equate to.

Understanding these expressions is critical because they form the basis for forming matrices, manipulating equations, and solving complex mathematical problems. They are the building blocks of more advanced topics in algebra, making them essential for not only solving systems but also for creating models of real-world situations.