Determine if the two equations are parallel, perpendicular, or the same line. If, perpendicular, at what point does the two lines intersect?

Determine if the two equations are parallel, perpendicular, or the same line.  If, perpendicular, at what point does the two lines intersect?

\(\small{ \begin{array}{lrclll} (1) & 2x+3y = -2 \qquad \Rightarrow \qquad y &=& -\frac23 x - \frac23 & m_1 = -\frac23 & b_1 = -\frac23 \\ (2) & x+y= 0 \qquad \Rightarrow \qquad y &=& -x & m_2 = -1 & b_2 = 0 \\ \end{array} }\)

1. Parallel ?

Parallel, if  \(\small{m_1 = m_2 \text{ and } b_1 \ne b_2}\)

We have: \(\small{m_1 \ne m_2 \qquad -\frac23 \ne -1 } \qquad \Rightarrow {\color{red}parallel~ no} \)

2. The same line ?

The same line, if  \(\small{m_1 = m_2 \text{ and } b_1 = b_2}\) 

We have:

\(\small{m_1 \ne m_2 \qquad  -\frac23 \ne -1 } \qquad \Rightarrow {\color{red}the~ same~ line~ no}\)\(\small{b_1 \ne b_2 \qquad  -\frac23 \ne 0 } \qquad \Rightarrow {\color{red}the~ same~ line~ no}\) 

3. Perpendicular ?  

Perpendicular, if  \(\small{m_1 = -\frac{1}{m_2}}\)  

We have: \(\small{m_1 \ne -\frac{1}{m_2} \qquad  -\frac23 \ne -\frac{1}{-1} } \qquad \Rightarrow {\color{red}perpendicular~ no}\)