91Ó°ÊÓ

A system of equations involves multiple equations that all share common variables. The main objective is to find the values of these variables that make all the equations true simultaneously.
When we solve a system, we are looking for these common values, also known as the 'solution set.'
For instance, in our exercise, we have the following system:
\(3x + 2y = 4\)
\(6x + 4y = 8\)
To better manage our equations, we can convert the system into a different format, like matrix form, which makes it easier to apply mathematical methods for solving.

Matrix form is a concise and organized way to represent a system of equations.
In our example, the system:
\(3x + 2y = 4\)
\(6x + 4y = 8\)
can be written as a matrix equation: \[ \begin{pmatrix} 3 & 2 \ 6 & 4 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 4 \ 8 \end{pmatrix} \]
This matrix equation has several components:

• The coefficient matrix \( \begin{pmatrix} 3 & 2 \ 6 & 4 \end{pmatrix} \) containing the coefficients of the variables
• The variable matrix \( \begin{pmatrix} x \ y \end{pmatrix} \)
• The constant matrix \( \begin{pmatrix} 4 \ 8 \end{pmatrix} \)

This form helps us to easily apply Cramer's Rule, or any other method, to solve for the variables.

Cramer's Rule provides a straightforward method for solving systems of linear equations using determinants. Let's outline the steps:

• First, compute the determinant of the coefficient matrix (let's call it \( D \)).
• Then, solve for each variable by replacing the respective column in the coefficient matrix with the constant matrix and compute the determinant of this new matrix.
• Divide the determinant of the new matrix by \( D \) to get the value of the variable.

In our exercise, if \( D = 0 \), Cramer's Rule cannot be applied because the system might be either dependent or inconsistent. If we compute \[ D = det \begin{pmatrix} 3 & 2 \ 6 & 4 \end{pmatrix} = 3 \times 4 - 6 \times 2 = 12 - 12 = 0 \] Thus, \ D \ is zero, indicating that this system doesn't have a unique solution. Instead, another method can be used to determine if the solution is dependent (infinitely many solutions) or inconsistent (no solution). For our system, both equations are multiples of each other, signaling infinitely many solutions along the line \(y = -1.5x + 2\).