Problem 10 - A Graphical Approach to College Algebra

Question

Write the augmented matrix for each system. Do not solve the system. $$\begin{aligned} 2 x+7 y &=1 \\ 5 x &=-15 \end{aligned}$$

Step-by-Step Solution

Answer

The augmented matrix is \( \begin{bmatrix} 2 & 7 & | & 1 \\ 5 & 0 & | & -15 \end{bmatrix} \).

Step 1: Identify the coefficients in the equations

The given system of equations is \( 2x + 7y = 1 \) and \( 5x = -15 \). First, identify the coefficients of the variables \( x \) and \( y \) in each equation. In the first equation, the coefficients are \( 2 \) for \( x \) and \( 7 \) for \( y \), with a constant of \( 1 \). In the second equation, the coefficient is \( 5 \) for \( x \), \( 0 \) for \( y \) (since \( y \) does not appear), and a constant of \( -15 \).

Step 2: Write the augmented matrix

Using the coefficients and constants identified in the first step, construct the augmented matrix. The coefficients of the variables \( x \) and \( y \) create the left part of the matrix, while the constants form the right part. The augmented matrix is written as: \[\begin{bmatrix}2 & 7 & | & 1 \5 & 0 & | & -15\end{bmatrix}\] This matrix represents the system of equations with each row corresponding to an equation.

Key Concepts

Systems of Equations

When working with systems of equations, you're dealing with a collection of two or more equations that have multiple variables. The main goal is often to find a solution that works for all equations in the system simultaneously.
To better understand this concept, imagine each equation as a rule that the variables must follow. The challenge is to find a set of variable values that satisfy all these rules at once.

• For example, in the system provided, the equations are:
• \(2x + 7y = 1\)
• \(5x = -15\)

This specific system is made up of two equations with two variables: \(x\) and \(y\). Instead of solving these here, we focus on representing them with matrices, which can reveal insightful properties of the system.

Coefficients

Coefficients are the numerical factors that multiply the variables in an equation. In the system of equations context, each coefficient tells you how much a certain variable contributes to the equation.
For example, in the equation \(2x + 7y = 1\), the number \(2\) is the coefficient of \(x\), and \(7\) is the coefficient of \(y\). These coefficients are crucial because they will fill up the matrix used to represent the equations.

• The coefficient of a missing variable is \(0\). In the second equation, \(5x = -15\), the coefficient for \(x\) is \(5\), and for \(y\) it is \(0\), since \(y\) is absent.

Recognizing coefficients helps to construct a matrix that aptly represents each equation in the system.

Matrix Representation

A matrix is a compact way to organize the coefficients and constants from a system of equations. In a matrix, rows represent individual equations, while columns correspond to their respective coefficients and constants.
When using matrices for systems of equations, an augmented matrix is often used, combining both the coefficients and the constants into a single matrix.

• From the given system \(2x + 7y = 1\) and \(5x = -15\), the augmented matrix becomes:\[\begin{bmatrix}2 & 7 & | & 1 \5 & 0 & | & -15\end{bmatrix}\]

This matrix aligns the coefficients of \(x\) and \(y\) in columns, with a vertical line separating them from the constants. It simplifies visualizing the entire system and is commonly used for further solving techniques like row reduction.

Linear Equations

Linear equations are equations where each term is either a constant or the product of a constant and a single variable. These equations represent straight lines when plotted on a graph.
In a system like \(2x + 7y = 1\), both terms involving \(x\) and \(y\) individually contribute to forming a line in a two-dimensional space. Linear equations express relationships where the change in one variable directly affects another, maintaining consistent rate increases or decreases.

• When these equations form a system, their intersection or lack thereof gives valuable insight into its solution, if it exists. This is why linear equations are foundational in algebra, offering a straightforward groundwork from which more complex systems and representations, like matrices, can be explored.

Understanding the linear aspect ensures success when approaching systems represented in various formats like matrices.