What is the equation of the line that is perpendicular to and has the same [tex]\( y \)[/tex]-intercept as the given line?

A. [tex]\( y = \frac{1}{5} x + 1 \)[/tex]
B. [tex]\( y = \frac{1}{5} x + 5 \)[/tex]
C. [tex]\( y = 5x + 1 \)[/tex]
D. [tex]\( y = 5x + 5 \)[/tex]



Answer :

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].