independent system
Has exactly one solution. The graph consists of two intersecting lines.
An independent system refers to a system of linear equations where there is a unique solution for each variable in the equations. In other words, an independent system has no redundant equations and no linearly dependent equations.
A system of linear equations is considered independent if its determinant is not equal to zero. The determinant is a mathematical concept that is calculated based on the coefficients of the equation. If the determinant is zero, it means that the equations are linearly dependent, and there are infinite solutions or no solutions to the system.
For example, consider the following system of equations:
2x + 3y = 6
4x + 6y = 12
To determine if this system is independent, we need to calculate the determinant of the coefficients. The determinant of this 2×2 matrix is:
(2 3)
(4 6)
D = (2 * 6) – (4 * 3) = 0
Since the determinant is zero, this means that the equations are linearly dependent, and there are infinite solutions to the system. This system is not independent.
However, if we consider the following system of equations:
3x – 2y = 5
4x + 7y = 1
The determinant of this 2×2 matrix is:
(3 -2)
(4 7)
D = (3 * 7) – (4 * -2) = 29
Since the determinant is nonzero, this means that the system is independent, and there is a unique solution for each variable in the equations.
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