Problem 16 - A Graphical Approach to College Algebra

Question

Write the system of equations associated with each augmented matrix. $$\left[\begin{array}{rr|r} 1 & -5 & -18 \\ 6 & 2 & 20 \end{array}\right]$$

Step-by-Step Solution

Answer

\(\begin{cases} x - 5y = -18 \\ 6x + 2y = 20 \end{cases}\)

Step 1: Identify the Variables

Let's consider the variables in our system of equations to be \(x\) and \(y\). The augmented matrix represents coefficients of these variables in the equations.

Step 2: Analyze the First Row

The first row of the matrix is \([1, -5 | -18]\). This can be interpreted as the equation: \(1x - 5y = -18\).

Step 3: Analyze the Second Row

The second row of the matrix is \([6, 2 | 20]\). This translates into the equation: \(6x + 2y = 20\).

Step 4: Write the System of Equations

By interpreting each row as a separate equation, the system of equations becomes: \(\begin{cases} x - 5y = -18 \ 6x + 2y = 20 \end{cases}\).

Key Concepts

System of Equations

A system of equations involves finding values for variables that satisfy multiple equations simultaneously. In the provided exercise, we have two equations which are composed of two unknowns often denoted as \(x\) and \(y\). These equations can be written as:

• \(x - 5y = -18\)
• \(6x + 2y = 20\)

Each equation represents a line on a two-dimensional graph. The solutions to the system occur where these lines intersect, i.e., they have common points. Understanding systems of equations is fundamental because it lays the groundwork for more advanced concepts in algebra, providing tools to analyze and solve real-world problems where variables are interdependent.

Matrix Representation

An augmented matrix is a compact, organized way to represent a system of equations using only the coefficients and constants. In our case, the matrix: \[ \left[\begin{array}{rr|r} 1 & -5 & -18 \ 6 & 2 & 20 \end{array}\right] \] represents the system of equations with a clear layout:

• The first two columns indicate the coefficients for the variables \(x\) and \(y\).
• The third column, separated by a vertical line, represents the constants on the right-hand side of each equation.

This format is particularly useful because it allows for the application of various algebraic techniques, like Gaussian elimination, to solve for the unknowns directly without writing out all the variable terms repeatedly. It's a foundational method in linear algebra for systematically solving complex systems.

Linear Algebra

Linear Algebra is the branch of mathematics focusing on vector spaces and linear mappings between them. It encompasses various concepts including matrices, systems of linear equations, and their applications. One of the main goals of linear algebra is to provide efficient methods for solving systems of equations like the one in our exercise. Augmented matrices are one of these methods, serving as a tool to transition from the equation format to a matrix format. This transition allows the use of advanced techniques such as row reduction and matrix determinants to identify solutions more swiftly and efficiently. Not only does linear algebra apply to theoretical math, but it also aids in solving real-world problems from engineering to economics, by modeling relationships and changes through linear systems.