To determine the relationship between the two lines given by the equations [tex]\( x + y = 2 \)[/tex] and [tex]\( y = x + 4 \)[/tex], we need to analyze their slopes.
1. Convert each equation to the slope-intercept form [tex]\( y = mx + b \)[/tex]
- For the first equation [tex]\( x + y = 2 \)[/tex]:
[tex]\[ y = -x + 2 \][/tex]
Here, the slope [tex]\(m_1\)[/tex] is [tex]\(-1\)[/tex].
- For the second equation [tex]\( y = x + 4 \)[/tex]:
This equation is already in the slope-intercept form where the slope [tex]\(m_2\)[/tex] is [tex]\(1\)[/tex].
2. Compare the slopes:
The slope of the first line is [tex]\( m_1 = -1 \)[/tex].
The slope of the second line is [tex]\( m_2 = 1 \)[/tex].
3. Determine the relationship between the slopes:
- Lines are parallel if their slopes are equal ([tex]\( m_1 = m_2 \)[/tex]).
- Lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex] ([tex]\( m_1 \times m_2 = -1 \)[/tex]).
- Lines are the same line if their slopes are equal and they have the same y-intercept.
- If none of these conditions hold, then the lines are not related in those ways.
Considering our calculated slopes:
[tex]\[ m_1 \times m_2 = (-1) \times 1 = -1 \][/tex]
Since the product of the slopes is [tex]\(-1\)[/tex], the lines are perpendicular.
Thus, the statement that explains the relationship between the lines [tex]\( x + y = 2 \)[/tex] and [tex]\( y = x + 4 \)[/tex] is:
They are perpendicular.
