Question

system a
line 1: $y = \frac{1}{3}x + 3$
line 2: $-x + 3y = 9$
this system of equations is:
inconsistent
consistent dependent
consistent independent
this means the system has:
a unique solution
solution:
no solution
infinitely many solutions
system b
line 1: $y = 2x + 3$
line 2: $y = -x + 3$
this system of equations is:
inconsistent
consistent dependent
consistent independent
this means the system has:
a unique solution
solution:
no solution
infinitely many solutions
system c
line 1: $y = \frac{3}{2}x + 1$
line 2: $y = \frac{3}{2}x - 4$
this system of equations is:
inconsistent
consistent dependent
consistent independent
this means the system has:
a unique solution
solution:
no solution
infinitely many solutions

Explanation:

System A

Step1: Rewrite line 2 in slope - intercept form

For line 2: \(-x + 3y=9\), we can solve for \(y\). Add \(x\) to both sides: \(3y=x + 9\), then divide by \(3\): \(y=\frac{1}{3}x+3\).

Step2: Analyze the system

Since line 1 (\(y=\frac{1}{3}x + 3\)) and line 2 (\(y=\frac{1}{3}x+3\)) are the same equation, the system is consistent dependent. This means the system has infinitely many solutions.

System B

Step1: Set the equations equal to each other

Set \(2x+3=-x + 3\).

Step2: Solve for \(x\)

Add \(x\) to both sides: \(2x+x+3=-x+x + 3\), so \(3x+3 = 3\). Subtract \(3\) from both sides: \(3x=0\), then \(x = 0\).

Step3: Solve for \(y\)

Substitute \(x = 0\) into \(y=2x+3\) (or \(y=-x + 3\)). We get \(y=3\).
Since the two lines intersect at a single point \((0,3)\), the system is consistent independent and has a unique solution.

System C

Step1: Set the equations equal to each other

Set \(\frac{3}{2}x+1=\frac{3}{2}x-4\).

Step2: Analyze the equation

Subtract \(\frac{3}{2}x\) from both sides: \(1=-4\), which is a contradiction.
Since the two lines are parallel (same slope \(\frac{3}{2}\)) and have different \(y\) - intercepts (\(1\) and \(-4\)), the system is inconsistent and has no solution.

Answer:

• System A: consistent dependent, infinitely many solutions.
• System B: consistent independent, a unique solution \((0,3)\).
• System C: inconsistent, no solution.