Question

system a
line 1: $y = -\frac{1}{3}x + 3$
line 2: $y = -\frac{1}{3}x - 3$
graph of two lines
this system of equations is:
\bigcirc inconsistent
\bigcirc consistent dependent
\bigcirc consistent independent
this means the system has:
\bigcirc a unique solution
solution: (\square, \square)
\bigcirc no solution
\bigcirc infinitely many solutions

Explanation:

Step1: Analyze slopes of lines

For Line 1: \( y = -\frac{1}{3}x + 3 \), slope \( m_1 = -\frac{1}{3} \).
For Line 2: \( y = -\frac{1}{3}x - 3 \), slope \( m_2 = -\frac{1}{3} \).
Since \( m_1 = m_2 \), lines are parallel.

Step2: Analyze y-intercepts

Line 1 y-intercept: \( 3 \); Line 2 y-intercept: \( -3 \).
Different y-intercepts, so lines never intersect.

Step3: Determine system type

Inconsistent systems have no solution (parallel lines, no intersection).
Consistent dependent: same line (infinitely many solutions).
Consistent independent: intersect at one point (unique solution).

Answer:

s:
This system of equations is: inconsistent
This means the system has: no solution
(For the solution box, since no solution, it remains empty as there's no (x,y) pair. But following the options, the first part: "inconsistent", second part: "no solution".)

First question answer: inconsistent
Second question answer: no solution