Answer:
Part 1) Is a inconsistent system
Part 2) Is a consistent independent system
Part 3) Is a consistent dependent system
see the attached figure
Step-by-step explanation:
we know that
If a system has at least one solution, it is said to be consistent .
If a consistent system has exactly one solution, it is independent .
If a consistent system has an infinite number of solutions, it is dependent
If a system not have solution, it is said to be inconsistent
Part 1) we have
[tex]x+5y=-2\\x+5y=4[/tex]
This system of linear equations has two parallel lines (their slopes are equal) with different y-intercept
so
The lines don't intersect
The system has no solution
therefore
Is a inconsistent system
Part 2) we have
[tex]y=3x+4\\-2x+y=4[/tex]
we know that
If two lines have different slopes, then the lines intersect at one point and the system of equations has one solution
In this problem, the lines have different slopes (m=3 and m=2)
so
The lines intersect at one point
The system has one solution
therefore
Is a consistent independent system
Part 3) we have
[tex]3x+y=4[/tex] ----> equation A
[tex]-6x-2y=-8[/tex] ----> equation B
Multiply equation A by -2 both sides
[tex]-2(3x+y)=-2(4)[/tex]
[tex]-6x-2y=-8[/tex] ----> equation C
Compare equation B and equation C
Are identical lines
so
The system has infinity solutions
therefore
Is a consistent dependent system
