To find the equation of the line that is perpendicular to [tex]\( y = -4x + 10 \)[/tex] and passes through the point [tex]\( (7, 2) \)[/tex], we can follow these steps:
1. Find the slope of the given line:
- The given line is [tex]\( y = -4x + 10 \)[/tex]. Here, the slope [tex]\( m_1 \)[/tex] is [tex]\(-4\)[/tex].
2. Determine the slope of the perpendicular line:
- Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]. Therefore, the slope of the perpendicular line [tex]\( m_2 \)[/tex] should satisfy [tex]\( m_1 \cdot m_2 = -1 \)[/tex].
- Given [tex]\( m_1 = -4 \)[/tex], we need to find [tex]\( m_2 \)[/tex] such that [tex]\( -4 \cdot m_2 = -1 \)[/tex].
- Solving for [tex]\( m_2 \)[/tex], we get [tex]\( m_2 = \frac{1}{4} \)[/tex].
3. Use the point-slope form to write the equation of the perpendicular line:
- The point-slope form of a line is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( (x_1, y_1) \)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope.
- Here, the point is [tex]\( (7, 2) \)[/tex] and the slope [tex]\( m_2 = \frac{1}{4} \)[/tex].
- Substituting these values into the point-slope form, we get:
[tex]\[
y - 2 = \frac{1}{4}(x - 7)
\][/tex]
4. Convert the equation to slope-intercept form:
- Simplify the equation to solve for [tex]\( y \)[/tex]:
[tex]\[
y - 2 = \frac{1}{4}x - \frac{1}{4} \cdot 7
\][/tex]
[tex]\[
y - 2 = \frac{1}{4}x - \frac{7}{4}
\][/tex]
[tex]\[
y = \frac{1}{4}x - \frac{7}{4} + 2
\][/tex]
[tex]\[
y = \frac{1}{4}x - \frac{7}{4} + \frac{8}{4}
\][/tex]
[tex]\[
y = \frac{1}{4}x + \frac{1}{4}
\][/tex]
Thus, the equation of the line that is perpendicular to [tex]\( y = -4x + 10 \)[/tex] and passes through the point [tex]\( (7, 2) \)[/tex] is [tex]\( y = \frac{x}{4} + \frac{1}{4} \)[/tex].
Therefore, the correct choice is:
[tex]\[ \boxed{y = \frac{x}{4} + \frac{1}{4}} \][/tex]
