91Ó°ÊÓ

Matrix representation is a technique used to simplify systems of linear equations. By organizing the coefficients of variables into a matrix, it becomes easier to apply mathematical methods for solving the system. For the given system of equations:

• Equation 1: \(x - 3y = -1\)
• Equation 2: \(4x + 3y = 11\)

We can represent this system in matrix form as:\[\begin{bmatrix}1 & -3 & | & -1\4 & 3 & | & 11\end{bmatrix}\]Here, the numbers before the variables \(x\) and \(y\) are placed into a matrix, with the constants on the right side forming an augmented column. Using a matrix allows us to apply systematic methods to find solutions.

Calculating the determinant is a crucial step in analyzing a system of linear equations. The determinant provides insight into the nature of the solutions of the system, especially if the system has two variables as in a 2x2 matrix:

• The formula for the determinant \(D\) of a matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is \(D = ad - bc\).

Let's calculate the determinant for our matrix:\[D = \begin{vmatrix} 1 & -3 \ 4 & 3 \end{vmatrix} = (1 \times 3) - (-3 \times 4) = 3 + 12 = 15\]Since the determinant \(D\) is not equal to zero, we know that the system of equations has a unique solution. A non-zero determinant confirms that the system has an exact and unambiguous solution.

A unique solution means there is a distinct set of values for the variables that satisfy all the equations in the system. When the determinant of the coefficient matrix is not zero, it ensures that the lines represented by the equations intersect at exactly one point. This unique point of intersection is our solution.
In our example, the determinant \(D\) is 15, indicating a unique solution exists. This occurs when both equations intersect at the single point \((x, y) = (2, 1)\). Understanding that the uniqueness of a solution is linked to a non-zero determinant helps in solving systems methodically and accurately.
Always remember, a unique solution signifies the equations form lines or planes that are neither parallel nor coincident.

The substitution method is a competent technique to find the solution of a system of linear equations after confirming the existence of a unique solution. Here’s how you use it:

• First, solve one of the equations for one variable in terms of the other. From our equations, we get \(x = 3y - 1\) from \(x - 3y = -1\).
• Next, substitute this expression for \(x\) into the other equation \(4x + 3y = 11\).

Now solve for \(y\):\[4(3y - 1) + 3y = 11\]Expand and simplify:\[12y - 4 + 3y = 11\]Combine like terms:\[15y = 15\]Divide by 15 to solve for \(y\):\[y = 1\]Substitute \(y\) back into the expression for \(x\):\[x = 3(1) - 1 = 2\]This methodical substitution grants us the solution, verifying our system means \((x, y) = (2, 1)\) is valid. The substitution method is especially helpful when systems have a clear and unique solution, as it reduces complexity and focuses directly on isolating variables.