Problem 5 - Algebra and Trigonometry Real Mathematics, Real People

Question

Refer to the system of linear equations \(\left\\{\begin{aligned}-2 x+3 y &=5 \\\ 6 x+7 y &=4 \end{aligned}\right.\). Is the augmented matrix for the system of dimension \(3 \times 2 ?\)

Step-by-Step Solution

Answer

No, the augmented matrix for the system of equations is of dimension \(2 \times 2\), not \(3 \times 2\).

Step 1: Identify the system

The system of equations is \(-2x + 3y = 5\) and \(6x + 7y = 4\). This is a system of two linear equations with two variables \(x\) and \(y\).

Step 2: Form the augmented matrix

To create the augmented matrix, write the coefficients of the variables into rows. For the first equation, the coefficients are -2 and 3. For the second equation, the coefficients are 6 and 7. This gives the following matrix: \[\begin{pmatrix}-2 & 3 \\ 6 & 7 \end{pmatrix}\] Note: In an augmented matrix, the column on the right of the bar (|) represents the constant terms of the equations. Since we don't include these in our count of the dimensions of the matrix, the bar and associated column is not represented here.

Step 3: Find the dimensions of the matrix

The dimension of a matrix is given by the number of rows times the number of columns. In this case, the matrix has 2 rows and 2 columns, so the dimension of the matrix is \(2 \times 2\).

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. In our original problem, we have two equations with two variables, \(x\) and \(y\):\

• \(-2x + 3y = 5\)
• \(6x + 7y = 4\)

When solving a system of equations, we aim to find values of \(x\) and \(y\) that satisfy all the equations involved. There are various methods for solving these systems, including substitution, elimination, and using matrices, which is the focus here.\
By converting these equations into matrix form, we can leverage the power of linear algebra techniques to simplify the problem-solving process.

Matrix Dimensions

Matrix dimensions refer to the number of rows and columns a matrix has. The matrix derived from our system of equations is usually an important visual and computational tool. In this case, the matrix is created from the coefficients of the system’s variables \((-2, 3)\) and \((6, 7)\):
\[\begin{pmatrix}-2 & 3 \6 & 7 \\end{pmatrix}\
\] To determine the dimensions of a matrix, we count the rows and columns it contains. For this matrix:

• There are 2 rows.
• There are 2 columns.

Thus, the matrix dimensions are \(2 \times 2\). Understanding the correct dimensions is crucial when performing matrix operations like addition, multiplication, or checking compatibility with other matrices.

Linear Algebra

Linear algebra is a branch of mathematics that focuses on vectors, matrices, and operations involving them. It provides tools for working with systems of linear equations and helps in understanding how solutions can be represented and manipulated.
In our exercise, linear algebra helps us use matrices to represent systems of equations more compactly. The concept of an augmented matrix is an extension of this tool, where we include an additional column representing the constants from the linear equations. This allows us to visualize and perform operations on equations as a whole.

• Matrices simplify complex systems into manageable forms.
• This helps apply systematic methods to solve them, such as reduction to row-echelon form or Gaussian elimination.

Understanding these processes requires grasping matrix operations and how they correspond to operations on equations. Linear algebra opens up powerful computational possibilities, making equation solving both efficient and scalable.