To find the equation of a line that is perpendicular to the given line [tex]\( y = -\frac{1}{4} x + 2 \)[/tex] and passes through the point [tex]\( (7, -3) \)[/tex], follow these steps:
1. Identify the slope of the given line:
- The given line is [tex]\( y = -\frac{1}{4} x + 2 \)[/tex].
- The slope of this line is [tex]\( -\frac{1}{4} \)[/tex].
2. Determine the slope of the perpendicular line:
- Perpendicular lines have slopes that are negative reciprocals of each other.
- The negative reciprocal of [tex]\( -\frac{1}{4} \)[/tex] is [tex]\( 4 \)[/tex].
- Therefore, the slope of the perpendicular line is [tex]\( 4 \)[/tex].
3. Use the point-slope form of a line equation:
- The point-slope form 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, we have the point [tex]\( (7, -3) \)[/tex] and the slope [tex]\( 4 \)[/tex].
- Plugging these values in, we get:
[tex]\[
y - (-3) = 4(x - 7)
\][/tex]
4. Simplify the equation to slope-intercept form:
- First, simplify inside the parentheses:
[tex]\[
y + 3 = 4(x - 7)
\][/tex]
- Distribute the slope:
[tex]\[
y + 3 = 4x - 28
\][/tex]
- Finally, solve for [tex]\( y \)[/tex] to put it in slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y = 4x - 28 - 3
\][/tex]
[tex]\[
y = 4x - 31
\][/tex]
Therefore, the equation of the line in slope-intercept form is:
[tex]\[
y = 4x - 31
\][/tex]
