To determine whether the given lines are parallel, perpendicular, or neither, we need to find the slopes of each line.
Starting with the first equation:
[tex]\[ x + y = 3 \][/tex]
To express this in slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y = -x + 3 \][/tex]
Here, [tex]\( m_1 = -1 \)[/tex] is the slope of the first line.
Now, consider the second equation:
[tex]\[ x - y = -3 \][/tex]
Rewriting this in slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ -y = -x - 3 \][/tex]
Multiplying both sides by -1 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = x + 3 \][/tex]
Here, [tex]\( m_2 = 1 \)[/tex] is the slope of the second line.
With the slopes [tex]\( m_1 = -1 \)[/tex] and [tex]\( m_2 = 1 \)[/tex], we can now determine their relationship:
1. Parallel Lines: If the slopes are equal. That is, [tex]\( m_1 = m_2 \)[/tex].
2. Perpendicular Lines: If the product of the slopes is [tex]\(-1\)[/tex]. That is, [tex]\( m_1 \times m_2 = -1 \)[/tex].
3. Neither: If the slopes do not satisfy either condition above.
Calculating the product of the slopes:
[tex]\[ m_1 \times m_2 = (-1) \times 1 = -1 \][/tex]
Since the product of the slopes [tex]\((-1) \times 1 = -1\)[/tex], the lines are perpendicular.
Hence, the correct answer is:
B. perpendicular
