To classify the system of equations:
[tex]\[
\begin{aligned}
3x + y - 1 & = 0 \quad \text{(Equation 1)} \\
\frac{1}{3}x + y - 6 & = 0 \quad \text{(Equation 2)}
\end{aligned}
\][/tex]
Step 1: Rearrange each equation to slope-intercept form [tex]\(y = mx + b\)[/tex]
Equation 1:
[tex]\[ 3x + y - 1 = 0 \][/tex]
[tex]\[ y = -3x + 1 \][/tex]
- Here, the slope [tex]\(m_1\)[/tex] is -3.
Equation 2:
[tex]\[ \frac{1}{3}x + y - 6 = 0 \][/tex]
[tex]\[ y = -\frac{1}{3}x + 6 \][/tex]
- Here, the slope [tex]\(m_2\)[/tex] is [tex]\(-\frac{1}{3}\)[/tex].
Step 2: Analyze the slopes [tex]\(m_1\)[/tex] and [tex]\(m_2\)[/tex]
- If the slopes are equal [tex]\(m_1 = m_2\)[/tex], the lines are either parallel or coincident.
- If the slopes are different [tex]\(m_1 \neq m_2\)[/tex], the lines are intersecting.
Here:
[tex]\[ m_1 = -3 \][/tex]
[tex]\[ m_2 = -\frac{1}{3} \][/tex]
Since [tex]\(-3 \neq -\frac{1}{3}\)[/tex], the slopes are different.
Step 3: Conclude the classification
With differing slopes, the lines represented by the equations are intersecting.
Thus, the correct answer is [tex]\(\boxed{\text{intersecting}}\)[/tex].
