To determine the relationship between the lines given by the equations [tex]\(2x + y = 4\)[/tex] and [tex]\(y = \frac{1}{2}x + 4\)[/tex], we follow these steps:
1. Convert the first equation to slope-intercept form: The equation of a line in the slope-intercept form is [tex]\( y = mx + b \)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
Given the equation [tex]\(2x + y = 4\)[/tex], we solve for [tex]\(y\)[/tex]:
[tex]\[
y = -2x + 4
\][/tex]
Therefore, the slope [tex]\(m_1\)[/tex] of the first line is [tex]\(-2\)[/tex].
2. Identify the slope of the second line: The second line is already in slope-intercept form [tex]\( y = \frac{1}{2}x + 4 \)[/tex].
So, the slope [tex]\(m_2\)[/tex] of the second line is [tex]\(\frac{1}{2}\)[/tex].
3. Compare the slopes to determine the relationship:
- Two lines are parallel if their slopes are equal.
- Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex], i.e., [tex]\(m_1 \times m_2 = -1\)[/tex].
- If neither of these conditions is met, then the lines are either the same line or not related.
In this case, the slopes are:
[tex]\[
m_1 = -2, \quad m_2 = \frac{1}{2}
\][/tex]
To check if they are perpendicular:
[tex]\[
m_1 \times m_2 = -2 \times \frac{1}{2} = -1
\][/tex]
Since the product of the slopes is [tex]\(-1\)[/tex], the lines are perpendicular.
Therefore, the lines [tex]\(2x + y = 4\)[/tex] and [tex]\(y = \frac{1}{2}x + 4\)[/tex] are perpendicular. The correct statement is:
They are perpendicular.
