To find the equation, in point-slope form, of the line that is perpendicular to the given line [tex]\( y + 5 = x + 2 \)[/tex] and passes through the point [tex]\((2, 5)\)[/tex], follow these steps:
1. Identify the slope of the given line:
- Start with the equation [tex]\( y + 5 = x + 2 \)[/tex].
- Rewrite it in slope-intercept form [tex]\( y = mx + b \)[/tex].
- Subtract 5 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = x - 3
\][/tex]
- Here, [tex]\( m = 1 \)[/tex] is the slope of the given line.
2. Determine the slope of the perpendicular line:
- The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope.
- The negative reciprocal of 1 is [tex]\(-1\)[/tex].
- Therefore, the slope of the perpendicular line is [tex]\(-1\)[/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.
- Substitute [tex]\( (x_1, y_1) = (2, 5) \)[/tex] and [tex]\( m = -1 \)[/tex]:
[tex]\[
y - 5 = -1(x - 2)
\][/tex]
4. Simplify the equation if necessary:
- Distribute the [tex]\(-1\)[/tex] on the right side:
[tex]\[
y - 5 = -x + 2
\][/tex]
- Add 5 to both sides to get it back in point-slope form (optional):
[tex]\[
y = -x + 7
\][/tex]
Hence, the equation in point-slope form of the line that is perpendicular to the given line and passes through the point [tex]\((2, 5)\)[/tex] is:
[tex]\[
y - 5 = -(x - 2)
\][/tex]
