To determine the equation, in point-slope form, of the line that is perpendicular to a given line and passes through a specific point, let's go through the process step-by-step.
1. Identify the slope of the given line:
- The given line is expressed as [tex]\( y + 5 = x + 2 \)[/tex].
- We need to convert this to the slope-intercept form [tex]\( y = mx + b \)[/tex] to identify the slope.
- Simplify the given equation:
[tex]\[
y + 5 = x + 2 \implies y = x + 2 - 5 \implies y = x - 3
\][/tex]
- Here, the coefficient of [tex]\( x \)[/tex] is the slope [tex]\( m \)[/tex]. Therefore, the slope of the given line is [tex]\( m = 1 \)[/tex].
2. Determine the slope of the perpendicular line:
- Perpendicular lines have slopes that are negative reciprocals of each other. If the slope of the given line is [tex]\( 1 \)[/tex], the slope [tex]\( m_2 \)[/tex] of the perpendicular line will be:
[tex]\[
m_2 = -\frac{1}{1} = -1
\][/tex]
3. Use the point-slope form of the equation:
- The point-slope form of a line's equation 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.
- Given the point [tex]\( (2, 5) \)[/tex] and the slope [tex]\( -1 \)[/tex], we substitute these values into the point-slope form:
[tex]\[
y - 5 = -1(x - 2)
\][/tex]
Thus, the equation of the line in point-slope form, which is perpendicular to the given line and passes through the point [tex]\((2, 5)\)[/tex], is:
[tex]\[
y - 5 = -1(x - 2)
\][/tex]
