Question

line 1: $y = 2x - 5$
line 2: $y = x - 4$
this system of equations is:
consistent independent
consistent dependent
inconsistent
this means the system has:
a unique solution
line 1: $y = -2x - 3$
line 2: $2x + y = -3$
this system of equations is:
consistent independent
consistent dependent
inconsistent
this means the system has:
a unique solution
line 1: $y = 2x + 2$
line 2: $y = 2x + 3$
this system of equations is:
consistent independent
consistent dependent
inconsistent
this means the system has:
a unique solution

Explanation:

Step1: Analyze the first system

For the first system \(y = 2x-5\) and \(y=x - 4\).
Set \(2x-5=x - 4\).
Subtract \(x\) from both sides: \(2x-x-5=x-x - 4\), so \(x-5=-4\).
Add \(5\) to both sides: \(x=-4 + 5=1\).
Substitute \(x = 1\) into \(y=x - 4\), \(y=1-4=-3\).
The two lines intersect at \((1,-3)\), so it is consistent independent (one - solution).

Step2: Analyze the second system

For the second system \(y=-2x - 3\) and \(2x+y=-3\).
Rewrite \(2x+y=-3\) as \(y=-2x - 3\).
The two equations are the same. So the system is consistent dependent (infinitely many solutions).

Step3: Analyze the third system

For the third system \(y = 2x+2\) and \(y=2x+3\).
Set \(2x+2=2x+3\).
Subtract \(2x\) from both sides: \(2x-2x+2=2x-2x+3\), \(2 = 3\) (a contradiction).
The two lines are parallel (same slope \(m = 2\), different \(y\) - intercepts). So the system is inconsistent (no solution).

Answer:

First system: consistent independent, a unique solution.
Second system: consistent dependent.
Third system: inconsistent.