Sure! To find the equation of the line parallel to [tex]\(3x + 2y = 8\)[/tex] and passing through the point [tex]\((-2, 5)\)[/tex], follow these steps:
1. Identify the slope of the given line:
- The given line is in the form [tex]\(3x + 2y = 8\)[/tex]. To find the slope, rewrite it in the slope-intercept form [tex]\(y = mx + c\)[/tex].
- Rearrange [tex]\(3x + 2y = 8\)[/tex] into the form [tex]\(y = mx + c\)[/tex]:
[tex]\[
2y = -3x + 8
\][/tex]
[tex]\[
y = -\frac{3}{2}x + 4
\][/tex]
- The slope [tex]\(m\)[/tex] of the given line is [tex]\(-\frac{3}{2}\)[/tex].
2. Use the point-slope form to find the equation of the new line:
- A line parallel to the given line will have the same slope. Therefore, the slope [tex]\(m\)[/tex] of our new line is also [tex]\(-\frac{3}{2}\)[/tex].
- The point-slope form of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
- Substitute the slope [tex]\(m = -\frac{3}{2}\)[/tex] and the point [tex]\((-2, 5)\)[/tex]:
[tex]\[
y - 5 = -\frac{3}{2}(x - (-2))
\][/tex]
[tex]\[
y - 5 = -\frac{3}{2}(x + 2)
\][/tex]
3. Simplify to the slope-intercept form:
- Distribute the slope [tex]\(-\frac{3}{2}\)[/tex]:
[tex]\[
y - 5 = -\frac{3}{2}x - 3
\][/tex]
- Solve for [tex]\(y\)[/tex]:
[tex]\[
y = -\frac{3}{2}x - 3 + 5
\][/tex]
[tex]\[
y = -\frac{3}{2}x + 2
\][/tex]
Therefore, the equation of the line parallel to [tex]\(3x + 2y = 8\)[/tex] and passing through the point [tex]\((-2, 5)\)[/tex] is [tex]\(y = -\frac{3}{2}x + 2\)[/tex].
Now, choose the appropriate values in the blanks:
[tex]\[
y = \boxed{-1.5}x + \boxed{2}
\][/tex]
