91Ó°ÊÓ

A system of linear equations is a set of two or more linear equations with the same variables. Linear equations can be represented in the form: \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. When dealing with such systems, the goal is to find the values of the variables that satisfy all equations simultaneously. Systems can have one unique solution, infinitely many solutions, or no solution at all. In our exercise, we have the system:
\[ \begin{cases} 1.5x + 3y = 5 \ 2x + 4y = 3 \end{cases} \]
We will use different methods to determine the solution, starting with Cramer's rule and then potentially using substitution or elimination methods.

The determinant is a special number calculated from a matrix. It's essential in solving systems of linear equations, especially using Cramer's rule. For a 2x2 matrix \ A \, the determinant is calculated as follows: \[ \text{det}(A) = \begin{vmatrix} a & b \ c & d \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = (ad - bc) \]
For the given system:\[ A = \begin{pmatrix} 1.5 & 3 \ 2 & 4 \ \ \ \end{pmatrix} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{pmatrix} \]
The determinant is computed as:
\[ D = (1.5 \ * 4 - 3 * 2) = 6 - 6 = 0 \ \ \ .\] Since D = 0, Cramer's rule cannot be used, and we need to use other methods.

The substitution method involves solving one of the equations for one variable in terms of the other variable and then substituting that expression into the other equation. This method is particularly useful when one equation is easy to solve for one of the variables.
From the first equation:
\[ 1.5x + 3y = 5 \ \ \ \ \ \ \ \ x = 5 - 3y \ \ , where 3y is subtracted from both sides and subsequently divided by 1.5\ \ \]
Substituting \( x = \ \ 5 - 3y/1.5 \ \) into the second equation:
\[ 2 * (5 - 3y/1.5) + 4y = 3 \ or simplified as:
\frac{10 - 6y + 4y}{1.5} = 3 = 10 - (4.5) 10 = (1.5)9 = 9\ \ x\]
As the results from all calculations indicate the system is inconsistent suggesting no solutions exist for overlapping system \

The elimination method involves adding or subtracting the equations in a system to eliminate one of the variables. This method is beneficial when both equations have a term that can be easily canceled out.
For instance, in our system:
\[ \begin{cases}1 1.5x + 3y = 5 \ 2x + 4y=3 = -1 + \ \end{cases} \ $\]
You can multiply the first equation by 2/3 eliminating (common term y\ \ \ \ \ \ \ \ \ both x\ \ \ results\
)\ \ \ \ \ half eliminated