Answer:
<em>Consistent independent</em>
<em>Consistent dependent</em>
<em>Inconsistent</em>
Step-by-step explanation:
<u>Consistent and Inconsistent Linear Systems</u>
A system of equations is consistent if it has at least one solution. If the solution is unique, then the system is consistent independent, otherwise is consistent dependent, i.e., it has an infinite number of solutions.
When there is no solution for the system, the lines of each equation are parallel but not coincident. A consistent dependent has two non-parallel lines, and a consistent independent system has only one line, i.e. the same line represents both equations.
Two lines are parallel if they have the same slope. To find the slopes, we represent the equations in slope-intercept form:
y = mx + b
Where m is the slope and b is the y-intercept.
Consider the system:
4x + y = 8
x + 3y = 8
Rearranging both equations:
y = -4x + 8
y = -1/3 x + 8
The slopes of the lines are -4 and -1/3 respectively, since they are different, the system is consistent independent.
Consider the system:
-4x + 6y = -2
2x - 3y = 1
Rearranging both equations:
y = 4/6 x - 2/6 = 2/3 x - 1/3
y = 2/3 x - 1/3
Both equations are exactly the same, thus the system is consistent dependent.
Consider the system:
5x - 2y = 4
5x - 2y = 6
Rearranging both equations:
y = 5/2 x - 2
y = 5/2 x - 3
These lines are parallel but not coincident. Thus this system is inconsistent.
