Problem 9 - A Graphical Approach to College Algebra

Question

Write the augmented matrix for each system. Do not solve the system. $$\begin{aligned} x+5 y &=6 \\ x &=3 \end{aligned}$$

Step-by-Step Solution

Answer

The augmented matrix is \(\begin{bmatrix} 1 & 5 & | & 6 \\ 1 & 0 & | & 3 \end{bmatrix}\).

Step 1: Identify the Coefficients

In the system of equations given, identify the coefficients of each variable and the constant on the right side of the equation. The first equation is \(x + 5y = 6\), with coefficients 1 (for \(x\)), 5 (for \(y\)), and a constant 6. The second equation is \(x = 3\), with coefficients 1 (for \(x\)), 0 (since there is no \(y\)), and a constant 3.

Step 2: Write the Augmented Matrix

The augmented matrix for a system of equations is constructed by listing the coefficients of the variables and constants in each equation in a matrix format. For the system \(x + 5y = 6\) and \(x = 3\), the augmented matrix is:\[ \begin{bmatrix} 1 & 5 & | & 6 \ 1 & 0 & | & 3 \end{bmatrix} \]The vertical bar (|) separates the coefficients of the variables from the constants.

Key Concepts

Systems of Equations

A system of equations consists of two or more equations with the same set of variables. The goal is often to find values for these variables that satisfy all equations simultaneously. In our example, the system is:

• \(x + 5y = 6\)
• \(x = 3\)

This system has two equations and involves two variables, \(x\) and \(y\). Breaking the set of equations can give insights into how variables relate to one another.
Not every system will have a simple solution. Some systems might not have any solution at all, while others, like this one, may have exactly one solution due to the constraints being represented.
This competence to express mathematical relationships using multiple equations comes handy in various fields such as engineering, economics, and physics.

Matrix Representation

Matrices can be used to represent systems of equations in a compact form. This involves writing the coefficients and constants in a structured, rectangular grid known as a matrix.
For our system \(x + 5y = 6\) and \(x = 3\), the matrix form will include coefficients along with constants divided by a vertical bar such as:

• The first row corresponding to \(x + 5y = 6\) becomes \([1, 5 | 6]\).
• The second row corresponding to \(x = 3\) becomes \([1, 0 | 3]\).
The beauty of matrix representation lies in its simplicity and uniform structure which is essential when utilizing computers to solve complex systems with many variables.

Coefficients in Algebra

To efficiently convert equations to matrix form, understanding coefficients is crucial. Coefficients are the numbers that multiply the variables in equations. In our equation \(x + 5y = 6\), the coefficient of \(x\) is 1 and that of \(y\) is 5.
In the second equation, \(x = 3\), the coefficient of \(x\) is 1. Since no \(y\) is present, its coefficient is 0. The ability to decipher coefficients correctly is core to forming the right augmented matrix.

• Presence: Indicates the participation of a variable in the equation.
• Magnitude: Suggests the impact or contribution of that variable.

This step aids in reducing emotional errors during matrix formation. Knowing how coefficients interact with variables is foundational for understanding complex algebraic concepts.

Linear Algebra

Linear algebra provides the framework for representing and solving systems using matrices. It is all about finding unknowns that will satisfy given relationships within linear systems.
In linear algebra, the matrix form can simplify and unify the expression of complex systems. It allows for algorithmic manipulation, making it simpler to apply transformations and solve equations.

• Efficiency: Solving large systems becomes feasible.
• Structure: Provides a consistent format for manipulation.

The transformation from equations to matrix form exemplifies a stepping stone from detailed manipulation of individual equations to using broader, efficient techniques. Understanding these fundamentals is invaluable, especially as you delve into computational solutions and broader applications such as machine learning and data science.