Problem 59 - Intermediate Algebra

Question

For the system \(\left\\{\begin{array}{l}2 x-3 y=5 \\ 4 x+8=y\end{array}\right.\), explain what is wrong with writing its corresponding augmented matrix as \(\left[\begin{array}{rrr}2 & -3 & 5 \\ 4 & 8 & 1\end{array}\right]\) How should it be written?

Step-by-Step Solution

Answer

The augmented matrix should be \( \left[ \begin{array}{rrr} 2 & -3 & 5 \\ 4 & -1 & -8 \end{array} \right] \). The error was in the second row with incorrect coefficients.

Step 1: Identify Equations from the System

The given system of equations is \( \begin{cases} 2x - 3y = 5 \ 4x + 8 = y \end{cases} \). Notice that the second equation is not in the standard form (\( ax + by = c \)). Let's rewrite it in standard form.

Step 2: Rearrange the Second Equation

Rewrite the second equation \( 4x + 8 = y \) as \( 4x - y = -8 \). The new system of equations becomes \( \begin{cases} 2x - 3y = 5 \ 4x - y = -8 \end{cases} \).

Step 3: Construct the Correct Augmented Matrix

Now both equations are in the standard form. Construct the augmented matrix as follows: \( \left[ \begin{array}{rr|r} 2 & -3 & 5 \ 4 & -1 & -8 \end{array} \right] \). This matrix correctly represents the system of equations.

Step 4: Compare the Given and Correct Augmented Matrices

The given matrix was \( \left[ \begin{array}{rrr} 2 & -3 & 5 \ 4 & 8 & 1 \end{array} \right] \). The issue is with the second row, where the coefficient of \( y \) should be \(-1\) and the constant should be \(-8\), not \(8\) and \(1\). The correct matrix is \( \left[ \begin{array}{rrr} 2 & -3 & 5 \ 4 & -1 & -8 \end{array} \right] \).

Key Concepts

Linear System of Equations

A linear system of equations is a collection of one or more linear equations involving the same set of variables. In the given exercise, we have two equations: \( 2x - 3y = 5 \) and \( 4x + 8 = y \). A solution to this system is a set of values for the variables that makes all equations true simultaneously.

To solve such systems, different methods can be applied, such as substitution, elimination, or using matrices. The goal is to find the point(s) at which all these linear equations intersect. For the equations to be in a usable form, consistency in their arrangement is crucial.

Understanding the form and manipulation of each equation is essential to correctly represent and eventually solve the linear system.

Matrix Representation

Matrix representation is a powerful tool to organize and manipulate linear equations, especially when dealing with multiple equations. In our exercise, once the linear system is rearranged into a standard form, it transforms the system into a matrix format, known as an augmented matrix.

An augmented matrix includes the coefficients of the variables and the constants on the right side of the equality. For our equations, it should be represented as:

• The first row for the first equation: coefficients \(2\) and \(-3\), and the constant \(5\).
• The second row for the rearranged second equation: coefficients \(4\) and \(-1\), and the constant \(-8\).

This matrix forms the basis for applying various algebraic techniques to find the solutions, including Gaussian elimination. Getting the correct matrix is key to solving the equations accurately.

Standard Form of a Linear Equation

The standard form of a linear equation is usually expressed as \( ax + by = c \) where \( a \), \( b \), and \( c \) are real numbers, and \( x \) and \( y \) are variables. This form is particularly useful for representing linear systems because it displays each term's coefficient clearly.In the exercise, we noticed an inconsistency with the second equation: \( 4x + 8 = y \). We rearranged it into its standard form: \( 4x - y = -8 \). This rearrangement ensures that each equation aligns correctly for matrix representation.Using the standard form helps maintain a consistent structure, essential when constructing matrices, allowing for easier comparisons and manipulations. By ensuring all equations in a system are in a standardized format, solving them becomes a streamlined process.