To classify the given system of equations:
[tex]\[ \begin{array}{l}
\frac{1}{3} x + y + 3 = 0 \\
\frac{1}{2} x + y + 2 = 0
\end{array} \][/tex]
we will follow these steps:
1. Express both equations in slope-intercept form:
- Convert [tex]\( \frac{1}{3} x + y + 3 = 0 \)[/tex] to [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y = -\frac{1}{3} x - 3 \][/tex]
- Convert [tex]\( \frac{1}{2} x + y + 2 = 0 \)[/tex] to [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y = -\frac{1}{2} x - 2 \][/tex]
2. Compare the slopes and intercepts:
- The first equation [tex]\( y = -\frac{1}{3} x - 3 \)[/tex] has a slope of [tex]\(-\frac{1}{3}\)[/tex].
- The second equation [tex]\( y = -\frac{1}{2} x - 2 \)[/tex] has a slope of [tex]\(-\frac{1}{2}\)[/tex].
3. Determine the relationship between the lines:
- If the slopes are equal and the intercepts are equal, the lines are coincident (the same line).
- If the slopes are equal but the intercepts are different, the lines are parallel.
- If the slopes are different, the lines are intersecting.
In this case, the slopes [tex]\(-\frac{1}{3}\)[/tex] and [tex]\(-\frac{1}{2}\)[/tex] are different.
Since the slopes are different, the lines must intersect at a single point.
Therefore, the system of equations is:
[tex]\[ \boxed{\text{intersecting}} \][/tex]
