To classify the system of equations, we'll follow these steps:
1. Analyze the given system of equations:
[tex]\[
\begin{cases}
2x + y - 4 = 0 \\
-6x + y + 6 = 0
\end{cases}
\][/tex]
2. Rewrite each equation in slope-intercept form ( [tex]\( y = mx + b \)[/tex] ) to identify the slopes and intercepts:
[tex]\[
\text{Equation 1:} \quad 2x + y - 4 = 0
\][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[
y = -2x + 4
\][/tex]
The slope ( [tex]\( m_1 \)[/tex] ) is [tex]\(-2\)[/tex] and the y-intercept ( [tex]\( b_1 \)[/tex] ) is [tex]\(4\)[/tex].
[tex]\[
\text{Equation 2:} \quad -6x + y + 6 = 0
\][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[
y = 6x - 6
\][/tex]
The slope ( [tex]\( m_2 \)[/tex] ) is [tex]\(6\)[/tex] and the y-intercept ( [tex]\( b_2 \)[/tex] ) is [tex]\(-6\)[/tex].
3. Compare the slopes and intercepts:
- The slopes [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] are [tex]\(-2\)[/tex] and [tex]\(6\)[/tex] respectively.
Since the slopes are different ([tex]\( m_1 \neq m_2 \)[/tex]), the lines are not parallel. Lines with different slopes will intersect at exactly one point.
4. Conclusion:
Given that the lines have different slopes, we can conclude that the system of equations is intersecting.
Thus, the correct answer is:
[tex]\[
\text{intersecting}
\][/tex]
