Sure, let's go through the steps to find the equation of the line parallel to the given equation [tex]\(3x + 2y = 8\)[/tex] and passing through the point [tex]\((-2, 5)\)[/tex].
1. Convert the given equation to slope-intercept form [tex]\(y = mx + b\)[/tex]:
Starting with the given equation:
[tex]\[
3x + 2y = 8
\][/tex]
Isolate [tex]\(y\)[/tex]:
[tex]\[
2y = -3x + 8
\][/tex]
[tex]\[
y = -\frac{3}{2}x + 4
\][/tex]
Hence, the slope ([tex]\(m\)[/tex]) of the given line is [tex]\(-\frac{3}{2}\)[/tex].
2. Determine the slope of the parallel line:
Lines that are parallel have the same slope. Therefore, the slope of the new line is also [tex]\(-\frac{3}{2}\)[/tex].
3. Use the point-slope form to find the equation of the parallel line:
The point-slope form of a line 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.
We are given the point [tex]\((-2, 5)\)[/tex] and the slope [tex]\(m = -\frac{3}{2}\)[/tex].
Substitute [tex]\( x_1 = -2 \)[/tex], [tex]\( y_1 = 5 \)[/tex], and [tex]\( m = -1.5\)[/tex] (which is [tex]\(-\frac{3}{2}\)[/tex]) into the point-slope form:
[tex]\[
y - 5 = -\frac{3}{2}(x + 2)
\][/tex]
4. Simplify to slope-intercept form ([tex]\(y = mx + b\)[/tex]):
Distribute the slope:
[tex]\[
y - 5 = -\frac{3}{2}x - 3
\][/tex]
Add 5 to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[
y = -\frac{3}{2}x - 3 + 5
\][/tex]
[tex]\[
y = -\frac{3}{2}x + 2
\][/tex]
Thus, the equation of the line parallel to [tex]\(3x + 2y = 8\)[/tex] that passes through the point [tex]\((-2, 5)\)[/tex] is:
[tex]\[
y = -\frac{3}{2} x + 2
\][/tex]
Select the correct answers from the drop-down menus:
The equation [tex]\(y =\)[/tex] [tex]\(\boxed{-\frac{3}{2}}\)[/tex] [tex]\( x + \boxed{2}\)[/tex] represents the line parallel to the given equation and passes through the point [tex]\((-2,5)\)[/tex].
