To determine the equation of the line parallel to the given line [tex]\(3x + 2y = 8\)[/tex] and passing through the point [tex]\((-2, 5)\)[/tex], we need to follow these steps:
1. Find the slope of the given line:
The given equation is [tex]\(3x + 2y = 8\)[/tex]. To find the slope, we first convert this equation to the slope-intercept form [tex]\(y = mx + b\)[/tex].
Solve for [tex]\(y\)[/tex]:
[tex]\[
2y = -3x + 8
\][/tex]
[tex]\[
y = -\frac{3}{2}x + 4
\][/tex]
Here, the slope [tex]\(m\)[/tex] is [tex]\(-\frac{3}{2}\)[/tex].
2. Form the equation of the parallel line:
A line parallel to the given line will have the same slope. Therefore, the slope of our desired line is also [tex]\(-\frac{3}{2}\)[/tex].
So, the equation of the parallel line in the slope-intercept form is:
[tex]\[
y = -\frac{3}{2}x + b
\][/tex]
3. Determine the y-intercept [tex]\(b\)[/tex]:
To find the y-intercept [tex]\(b\)[/tex], use the point [tex]\((-2, 5)\)[/tex] that the line passes through. Substitute [tex]\(x = -2\)[/tex] and [tex]\(y = 5\)[/tex] into the equation:
[tex]\[
5 = -\frac{3}{2}(-2) + b
\][/tex]
Calculate it step-by-step:
[tex]\[
5 = 3 + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
b = 5 - 3
\][/tex]
[tex]\[
b = 2
\][/tex]
4. Write the final equation:
Now that we have both the slope and the y-intercept, 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]
Thus, the correct answers to the blanks are:
- Slope: [tex]\(-\frac{3}{2}\)[/tex]
- Y-intercept: [tex]\(2\)[/tex]
So 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 passing through the point [tex]\((-2,5)\)[/tex].
