To find the equation of the line that is perpendicular to [tex]\(3x + y = -5\)[/tex] and passes through the point [tex]\( (3, -7) \)[/tex], follow these steps:
1. Determine the slope of the given line:
The equation of the line is given in the form [tex]\(Ax + By = C\)[/tex]. For the line [tex]\(3x + y = -5\)[/tex], the coefficients are [tex]\(A = 3\)[/tex] and [tex]\(B = 1\)[/tex]. The slope (m) of a line in this form is given by [tex]\(-\frac{A}{B}\)[/tex].
[tex]\[
\text{slope} = -\frac{3}{1} = -3
\][/tex]
2. Find the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the original line’s slope. The negative reciprocal of [tex]\(-3\)[/tex] is [tex]\(\frac{1}{3}\)[/tex].
[tex]\[
\text{slope of the perpendicular line} = \frac{1}{3}
\][/tex]
3. Use the point-slope form to find the equation:
The point-slope form of a line's equation is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\(m\)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is a point on the line. Here, the slope [tex]\(m = \frac{1}{3}\)[/tex] and the point is [tex]\((3, -7)\)[/tex].
[tex]\[
y - (-7) = \frac{1}{3}(x - 3)
\][/tex]
Simplifying this:
[tex]\[
y + 7 = \frac{1}{3}(x - 3)
\][/tex]
4. Convert to standard form:
To eliminate the fraction, multiply both sides by 3:
[tex]\[
3(y + 7) = x - 3
\][/tex]
Simplify and rearrange to standard form [tex]\(Ax + By = C\)[/tex]:
[tex]\[
3y + 21 = x - 3
\][/tex]
Move [tex]\(x\)[/tex] to the left side of the equation:
[tex]\[
-x + 3y = -24
\][/tex]
Therefore, the equation of the line perpendicular to [tex]\(3x + y = -5\)[/tex] and passing through [tex]\((3, -7)\)[/tex] is:
[tex]\[
-x + 3y = -24
\][/tex]
