To find the equation of the line perpendicular to [tex]\(3x + y = -8\)[/tex] and passing through the point [tex]\((-3, 1)\)[/tex], follow these steps:
1. Find the slope of the original line:
The given line is [tex]\(3x + y = -8\)[/tex].
We need to rewrite this equation in slope-intercept form ([tex]\(y = mx + b\)[/tex]) to easily determine its slope.
[tex]\[
y = -3x - 8
\][/tex]
From this, we see the slope [tex]\(m\)[/tex] of the original line is [tex]\(-3\)[/tex].
2. Determine the slope of the perpendicular line:
The slope of any line perpendicular to another is the negative reciprocal of the original slope. Given the original slope [tex]\(m = -3\)[/tex], the slope [tex]\(m'\)[/tex] of the perpendicular line will be:
[tex]\[
m' = -\frac{1}{-3} = \frac{1}{3}
\][/tex]
3. Use the point-slope form to find the equation of the perpendicular line:
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.
Here, the point [tex]\((-3, 1)\)[/tex] lies on the perpendicular line, and the slope [tex]\(m' = \frac{1}{3}\)[/tex]. Substituting these values into the point-slope form equation:
[tex]\[
y - 1 = \frac{1}{3}(x + 3)
\][/tex]
4. Simplify to get the slope-intercept form:
We now solve for [tex]\(y\)[/tex] to convert the equation into slope-intercept form.
[tex]\[
y - 1 = \frac{1}{3}(x + 3)
\][/tex]
Distribute [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
y - 1 = \frac{1}{3}x + 1
\][/tex]
Add 1 to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[
y = \frac{1}{3}x + 2
\][/tex]
Thus, the equation of the line perpendicular to [tex]\(3x + y = -8\)[/tex] that passes through the point [tex]\((-3, 1)\)[/tex] is:
[tex]\[
\boxed{y = \frac{1}{3}x + 2}
\][/tex]
