Sure, let's solve this step by step.
1. Convert the given equation to slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ 3x + 2y = 8 \][/tex]
Solve for [tex]\( y \)[/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. Identify the slope of the parallel line:
For the line to be parallel, it must have the same slope as the given line, which is [tex]\( -\frac{3}{2} \)[/tex].
3. Determine the equation of the line passing through the point (-2, 5):
Use the point-slope form of the equation [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is the given point (-2, 5).
Substitute [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 the equation:
[tex]\[ y - 5 = -\frac{3}{2}x - 3 \][/tex]
Add 5 to both sides:
[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 passes through the point [tex]\((-2,5)\)[/tex].
Thus, the correct answers for the equation are:
- [tex]\( y = \boxed{-\frac{3}{2}} \)[/tex] [tex]\( x + \boxed{2} \)[/tex].
