To determine the equation of the line perpendicular to [tex]\(2x - 3y = 13\)[/tex] that passes through the point [tex]\((-6, 5)\)[/tex], follow these steps:
1. Find the slope of the original line [tex]\(2x - 3y = 13\)[/tex]:
- The standard form of the line is [tex]\(Ax + By = C\)[/tex].
- Rearrange this equation into slope-intercept form [tex]\(y = mx + b\)[/tex] to identify the slope [tex]\(m\)[/tex].
- Rewrite [tex]\(2x - 3y = 13\)[/tex] as:
[tex]\[
-3y = -2x + 13
\][/tex]
- Divide every term by -3 to solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{2}{3}x - \frac{13}{3}
\][/tex]
- Thus, the slope [tex]\(m\)[/tex] of the original line is [tex]\(\frac{2}{3}\)[/tex].
2. Determine the slope of the perpendicular line:
- The slope of a line perpendicular to another is the negative reciprocal of the original slope.
- The negative reciprocal of [tex]\(\frac{2}{3}\)[/tex] is:
[tex]\[
-\frac{3}{2}
\][/tex]
3. Find the equation of the new line using the point (-6, 5):
- Use the point-slope form of the equation: [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 [tex]\((-6, 5)\)[/tex].
- Substitute [tex]\(m = -\frac{3}{2}\)[/tex], [tex]\(x_1 = -6\)[/tex], and [tex]\(y_1 = 5\)[/tex]:
[tex]\[
y - 5 = -\frac{3}{2}(x + 6)
\][/tex]
- Simplify the equation:
[tex]\[
y - 5 = -\frac{3}{2}x - 9
\][/tex]
[tex]\[
y = -\frac{3}{2}x - 4
\][/tex]
Therefore, the equation of the line perpendicular to [tex]\(2x - 3y = 13\)[/tex] that passes through the point [tex]\((-6, 5)\)[/tex] is:
[tex]\[
y = -\frac{3}{2}x - 4
\][/tex]
Among the given choices, the correct equation is:
[tex]\[
y = -\frac{3}{2} x - 4
\][/tex]
