To find the equation of a line that passes through the point [tex]\((-3, 8)\)[/tex] and is parallel to the given line [tex]\( -5x - 6y = -18 \)[/tex], we need to follow several steps.
1. Determine the slope of the given line:
First, put the equation [tex]\(-5x - 6y = -18\)[/tex] into slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
[tex]\[
-5x - 6y = -18
\][/tex]
[tex]\[
-6y = 5x - 18
\][/tex]
[tex]\[
y = -\frac{5}{6}x + 3
\][/tex]
The slope ([tex]\(m\)[/tex]) of the given line is [tex]\(-\frac{5}{6}\)[/tex].
2. Identify the slope of the new line:
Since the new line is parallel to the given line, it has the same slope. Thus, the slope of the new line is also [tex]\(-\frac{5}{6}\)[/tex].
3. Use the point-slope form to find the equation of the new line:
The point-slope form of a line 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 is [tex]\((-3, 8)\)[/tex] and the slope is [tex]\(-\frac{5}{6}\)[/tex].
[tex]\[
y - 8 = -\frac{5}{6}(x + 3)
\][/tex]
Simplify this to obtain the slope-intercept form:
[tex]\[
y - 8 = -\frac{5}{6}x - \frac{15}{6}
\][/tex]
[tex]\[
y - 8 = -\frac{5}{6}x - 2.5
\][/tex]
[tex]\[
y = -\frac{5}{6}x + 5.5
\][/tex]
4. Convert the slope-intercept form to standard form [tex]\(Ax + By = C\)[/tex]:
[tex]\[
y = -\frac{5}{6}x + 5.5
\][/tex]
Multiply all terms by 6 to eliminate the fraction:
[tex]\[
6y = -5x + 33
\][/tex]
Rearrange to bring all terms to one side:
[tex]\[
5x + 6y = 33
\][/tex]
So, the equation of the line that goes through the point [tex]\((-3, 8)\)[/tex] and is parallel to the line [tex]\(-5x - 6y = -18\)[/tex] is:
[tex]\[
\boxed{5x + 6y = 33}
\][/tex]
