Classify the following system of equations: 8x - 12y = -9 18x + 27y = 21

To classify the system of equations, we need to look at the coefficients of the variables x and y in each equation. The first step is to write down the coefficients in an augmented matrix form:

\[ \begin{bmatrix} 8 & -12 \\ 18 & 27 \\ \end{bmatrix} \]

Next, we calculate the determinant (D) of the matrix formed by the coefficients of x and y. For a 2x2 matrix, the determinant is found by multiplying the top left element by the bottom right element and subtracting the product of the top right element by the bottom left element.

\[ D = (8)(27) - (-12)(18) \] \[ D = 216 - (-216) \] \[ D = 216 + 216 \] \[ D = 432 \]

Since the determinant D is non-zero (D ≠ 0), it implies that the system of equations is consistent and independent. This means there is exactly one solution to this system, and the lines represented by these equations intersect at a single point.

Extra: When you are dealing with a system of linear equations, particularly two equations with two variables, you have several possibilities for how the equations interact with each other.

1. Independent and Consistent: This occurs when you have a non-zero determinant (as in this case). The lines intersect at exactly one point, and there is a unique solution for x and y.

2. Dependent: This is when the two equations actually represent the same line. This would mean an infinite number of solutions because the lines lie on top of each other, covering every point on the line. In matrix terms, this would lead to a determinant of zero, and the equations would be scalar multiples of each other.

3. Inconsistent: This is when the two equations represent parallel lines that never intersect. In terms of solutions, this would mean there are no solutions, as the lines never meet. The determinant for such a system is also zero, but unlike the dependent system, the equations are not multiples of each other.

Ultimately, to classify systems like these, you look at the coefficients to determine the relationship between the two equations. In this particular example, because you end up with a non-zero determinant, you can confidently classify the system as independent and consistent, with exactly one solution.