Problem 33 - Algebra A Combined Function

Question

For the system \(\left\\{\begin{aligned} 2 x-3 y &=8 \\ x+5 y &=-3 \end{aligned}\right.\), explain what is wrong with writing the corresponding matrix as \(\left[\begin{array}{ccc}2 & 3 & 8 \\ 0 & 5 & -3\end{array}\right]\).

Step-by-Step Solution

Answer

The matrix is incorrect because the first row has the wrong sign for -3, the second row wrongly starts with 0, and it lacks a separator for the augmented part.

Step 1: Identify the System of Equations

The system given is \( \begin{aligned} 2x - 3y &= 8 \ x + 5y &= -3 \end{aligned} \). We need to represent this system in matrix form.

Step 2: Determine the Coefficients

For the equation \(2x - 3y = 8\), the coefficients are 2 and -3, while the constant is 8. For the equation \(x + 5y = -3\), the coefficients are 1 and 5, and the constant is -3.

Step 3: Construct the Correct Augmented Matrix

The correct augmented matrix based on the coefficients and constants is \( \left[ \begin{array}{cc|c} 2 & -3 & 8 \ 1 & 5 & -3 \end{array} \right] \). An augmented matrix includes both the coefficients and the constants separated by a vertical line.

Step 4: Find the Error in the Given Matrix

The provided matrix \( \left[ \begin{array}{ccc} 2 & 3 & 8 \ 0 & 5 & -3 \end{array} \right] \) contains errors. The first row should have a negative 3 instead of a 3. The second row incorrectly starts with a zero and should start with a 1, while the matrix given does not correctly use a format indicating it's an augmented matrix as there's no separator.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. In our example, the variables are "x" and "y."
The goal when working with a system like \(\begin{aligned}2x - 3y &= 8 \x + 5y &= -3\end{aligned} \)
is to find the values of "x" and "y" that satisfy both equations simultaneously. There are several methods to solve such systems, including substitution, elimination, and using matrix representation.
This means converting these algebraic equations into a form that can be handled more easily, such as using matrices.
Understanding systems of equations is crucial because they often represent real-world situations where different conditions interact with each other. Making sense of these interdependent relationships can help solve complex problems in arithmetic, physics, and engineering.

Matrix Representation

Matrix representation is a method of organizing a system of equations into an array of numbers called a matrix. Each row corresponds to one of the equations, and each column corresponds to one type of term: coefficients of "x," coefficients of "y," and constants.
For instance, in our system \(\begin{aligned} 2x - 3y &= 8 \ x + 5y &= -3 \end{aligned} \), the augmented matrix would be represented as:

• The first row represents the equation \(2x - 3y = 8\).
• The second row represents the equation \(x + 5y = -3\).

To construct this, we write a matrix:\[\begin{array}{cc|c}2 & -3 & 8 \1 & 5 & -3\end{array}\]Here, the vertical line separates the coefficients of the variables from the constants. This is crucial for distinguishing between the parts of each equation when using matrix operations like Gaussian elimination or applying the inverse matrix method to solve the system.

Coefficients in Matrices

Coefficients in a matrix come from the coefficients of the variables in the system of equations. They're an essential part of the matrix representation because they tell us the weight or influence of each variable in each equation.
In our exemplified system \(\begin{aligned} 2x - 3y &= 8 \ x + 5y &= -3 \end{aligned} \), the coefficients are:

• For the first equation, \(2x - 3y = 8\), the coefficients are 2 for "x" and -3 for "y."
• For the second equation, \(x + 5y = -3\), the coefficients are 1 for "x" and 5 for "y."

These numerical values are placed in their corresponding positions in the matrix:
- The first column holds coefficients of "x."- The second column holds coefficients of "y."
The correct positioning of coefficients is vital because any mistake can lead to incorrect solutions. Incorrectly writing coefficients is often the main issue students face when first learning about augmented matrices. Careful attention to sign and order ensures that the matrix properly represents the original system of equations.