Question

line 2: -3x + 2y = 0
this system of equations is:
consistent independent
consistent dependent
inconsistent
this means the system has:
a unique solution
solution:
infinitely many solutions
line 2: y = x - 5
this system of equations is:
consistent independent
consistent dependent
inconsistent
this means the system has:
a unique solution
solution:
infinitely many solutions
line 2: y = -2x + 4
this system of equations is:
consistent independent
consistent dependent
inconsistent
this means the system has:
a unique solution
solution:
infinitely many solutions

Explanation:

Step1: Analyze the first system

For the system \( - 3x + 2y=0\) (Line 2) and Line 1 (assuming it's a non - parallel line). A system is consistent dependent if the equations represent the same line. Here, they are not the same line. A system is inconsistent if the lines are parallel. Since the slopes are different (rewrite \( - 3x + 2y = 0\) as \(y=\frac{3}{2}x\), and if Line 1 has a different slope), it is consistent independent (unique solution).

Step2: Analyze the second system

For the system \(y=x - 5\) (Line 2) and Line 1. The slopes are different (assuming Line 1 has a slope not equal to 1). So it is consistent independent (unique solution).

Step3: Analyze the third system

For the system \(y=-2x + 4\) (Line 2) and Line 1. If the slopes of Line 1 and Line 2 are equal (both \(m=-2\)) and the y - intercepts are different, the lines are parallel. A system with parallel lines is inconsistent (no solution).

Answer:

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