Problem 12 - College Algebra with Corequisite Support

Question

For the following exercises, write the linear system from the augmented matrix. $$ \left[\begin{array}{rr|r} 3 & 4 & 10 \\ 10 & 17 & 439 \end{array}\right] $$

Step-by-Step Solution

Answer

The system is \(3x + 4y = 10\) and \(10x + 17y = 439\).

Step 1: Identify the Augmented Matrix

The provided augmented matrix is given as: \[\begin{bmatrix} 3 & 4 & 10 \ 10 & 17 & 439 \end{bmatrix}\]This matrix represents a system of linear equations.

Step 2: Assign Variables to the System

We'll assign variables to each column. Let's use \(x\) for the first column and \(y\) for the second column. This results in equations of the form: \[ax + by = c\] where \(a\) and \(b\) are coefficients and \(c\) is the constant.

Step 3: Write the First Equation

Using the first row of the augmented matrix, \(3x + 4y = 10\). The first row \([3 \underline{\phantom{xxx}} 4 \underline{\phantom{xxx}} | \underline{\phantom{xxx}} 10]\) represents this equation.

Step 4: Write the Second Equation

Using the second row of the augmented matrix, \(10x + 17y = 439\). The second row \([10 \underline{\phantom{xxx}} 17 \underline{\phantom{xxx}} | \underline{\phantom{xxx}} 439]\) represents this equation.

Key Concepts

Augmented Matrix

An augmented matrix is a very useful concept in linear algebra. It provides a way to represent a system of linear equations in a compact form. Let's break it down. An augmented matrix combines the coefficients of the variables on the left and the constants from the right side of the equations into one expanded matrix. This matrix serves as a concise summary of the entire system of equations.

Consider the matrix given in the exercise:

• The numbers in the first part of the matrix (before the vertical line) are the coefficients of the variables.
• The numbers after the vertical line represent the constants from each equation.

Understanding the layout of an augmented matrix is crucial because it allows you to directly translate between the matrix form and the system of equations it represents.

System of Linear Equations

A system of linear equations is a collection of one or more equations that you deal with altogether. Each equation in the system represents a line (or hyperplane in higher dimensions), and the solution to the system is the point or points where these lines intersect.

In a system of linear equations, each equation typically describes a relationship between two or more unknowns, which we call variables. The goal is to find the values of these variables that satisfy all the equations simultaneously.

• In mathematical terms:\[ ax + by = c \]Here, \(a\) and \(b\) are coefficients, \(x\) and \(y\) are variables, and \(c\) is the constant.
• Solving a system of linear equations often involves methods such as substitution, elimination, or the use of matrices.

In the exercise example, the system is comprised of two equations, and the solution would be where the lines represented by these equations intersect in the plane.

Variables Assignment

Variables assignment refers to the process of designating symbols, commonly such as \(x\), \(y\), or \(z\), to represent the unknown quantities in your system of equations. This step is crucial for translating a problem stated in real-world terms into a mathematical form.

In the context of matrices and linear equations:

• Each column before the vertical line in the augmented matrix corresponds to a variable.
• For instance, in our problem, the first column corresponds to the variable \(x\), and the second to \(y\).

By assigning variables correctly, you can rewrite the coefficients and constants from the augmented matrix into equations that can be solved to find the values of these unknowns.

Matrix Representation

Matrix representation is a way to express mathematical concepts, especially systems of equations, in a structured and manageable way by focusing on the coefficients and constants. This representation simplifies situations involving multiple linear equations.

With a matrix representation:

• The rows of the matrix correspond to the individual equations within a system.
• The columns correspond to the variables within the equations.
• By applying matrix operations, one can solve the entire system efficiently.

For example, the matrix\[\left[\begin{array}{rr|r}3 & 4 & 10 \10 & 17 & 439\end{array}\right]\]provides a way to visually interpret the system and apply various solution techniques, such as Gaussian elimination, to find the values of the variables involved.