To find the equation of a line that is perpendicular to a given line and has the same [tex]\( y \)[/tex]-intercept, follow these steps:
1. Identify the slope of the given line:
The equation of our given line is [tex]\( y = \frac{1}{5} x + 1 \)[/tex]. Here, the coefficient of [tex]\( x \)[/tex] is the slope, [tex]\( m \)[/tex], which is [tex]\( \frac{1}{5} \)[/tex].
2. Find the slope of the perpendicular line:
Two lines are perpendicular if the product of their slopes is [tex]\( -1 \)[/tex]. If the slope of the given line is [tex]\( \frac{1}{5} \)[/tex], the slope of the perpendicular line [tex]\( m_{\text{perpendicular}} \)[/tex] must satisfy:
[tex]\[
\left(\frac{1}{5}\right) \times m_{\text{perpendicular}} = -1
\][/tex]
To find [tex]\( m_{\text{perpendicular}} \)[/tex]:
[tex]\[
m_{\text{perpendicular}} = -1 \times \frac{5}{1} = -5
\][/tex]
Thus, the slope of the line perpendicular to our given line is [tex]\( -5 \)[/tex].
3. Use the same [tex]\( y \)[/tex]-intercept:
The given line has a [tex]\( y \)[/tex]-intercept of [tex]\( 1 \)[/tex]. A line that is perpendicular to our given line and has the same [tex]\( y \)[/tex]-intercept will also have [tex]\( y \)[/tex]-intercept [tex]\( 1 \)[/tex].
4. Form the equation of the perpendicular line:
Now that we have the slope [tex]\( -5 \)[/tex] and the [tex]\( y \)[/tex]-intercept [tex]\( 1 \)[/tex], we can write the equation in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept:
[tex]\[
y = -5x + 1
\][/tex]
So, the equation of the line that is perpendicular to the given line and has the same [tex]\( y \)[/tex]-intercept is:
[tex]\[
y = -5x + 1
\][/tex]
Hence, the correct answer is:
[tex]\( y = -5x + 1 \)[/tex].
