Sure! Let's determine the equation of the line that passes through point [tex]\((-3, 4)\)[/tex] and is perpendicular to a line with a slope of [tex]\(\frac{2}{5}\)[/tex].
1. Determine the slope of the perpendicular line:
- The slope of the given line is [tex]\(\frac{2}{5}\)[/tex].
- The slope of a perpendicular line is the negative reciprocal of [tex]\(\frac{2}{5}\)[/tex]. The negative reciprocal is [tex]\(-\frac{5}{2}\)[/tex].
2. Use the point-slope form of the equation of a line:
- The point-slope form is [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is the point the line passes through.
- Here, [tex]\((x_1, y_1) = (-3, 4)\)[/tex] and [tex]\(m = -\frac{5}{2}\)[/tex].
- Substituting these values into the point-slope form:
[tex]\[
y - 4 = -\frac{5}{2}(x + 3)
\][/tex]
3. Convert to standard form (Ax + By = C):
- First, eliminate the fraction by multiplying every term by 2:
[tex]\[
2(y - 4) = -5(x + 3)
\][/tex]
- Distribute the 2 and -5:
[tex]\[
2y - 8 = -5x - 15
\][/tex]
- Bring all terms to one side to write in the form [tex]\(Ax + By = C\)[/tex]:
[tex]\[
5x + 2y = 7
\][/tex]
Thus, the standard form of the equation of the line passing through [tex]\((-3, 4)\)[/tex] and perpendicular to the line with slope [tex]\(\frac{2}{5}\)[/tex] is:
[tex]\[
5x + 2y = 7
\][/tex]
