To solve for the equation of the line parallel to the given equation and passing through the point [tex]\((-2, 5)\)[/tex]:
1. Determine the slope of the given line.
We start with the given equation [tex]\(3x + 2y = 8\)[/tex]. To convert this into slope-intercept form ([tex]\(y = mx + b\)[/tex]):
[tex]\[
2y = -3x + 8
\][/tex]
[tex]\[
y = -\frac{3}{2}x + 4
\][/tex]
From this, we see that the slope [tex]\(m\)[/tex] of the given line is [tex]\(-\frac{3}{2}\)[/tex].
2. Determine the slope of the parallel line.
Since parallel lines have the same slope, the slope of the line parallel to the given line is also [tex]\(-\frac{3}{2}\)[/tex].
3. Use the point-slope form to find the equation of the new line.
The line must pass through the point [tex]\((-2, 5)\)[/tex]. Using the point-slope form:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substituting [tex]\(m = -\frac{3}{2}\)[/tex], [tex]\(x_1 = -2\)[/tex], and [tex]\(y_1 = 5\)[/tex]:
[tex]\[
y - 5 = -\frac{3}{2}(x + 2)
\][/tex]
4. Simplify to slope-intercept form.
Distribute and simplify:
[tex]\[
y - 5 = -\frac{3}{2}x - 3
\][/tex]
[tex]\[
y = -\frac{3}{2}x - 3 + 5
\][/tex]
[tex]\[
y = -\frac{3}{2}x + 2
\][/tex]
So, the equation [tex]\(y = -\frac{3}{2}x + 2\)[/tex] represents the line parallel to the given equation and passing through the point [tex]\((-2, 5)\)[/tex].
Therefore, the correct entries in the drop-down menus are:
- [tex]\( -\frac{3}{2} \)[/tex]
- [tex]\( 2 \)[/tex]
