To find the equation of the line that is parallel to the given line [tex]\(3x - 4y = -17\)[/tex] and passes through the point [tex]\((-3, 2)\)[/tex], we can follow these steps:
1. Identify the slope of the given line:
The given line equation is [tex]\(3x - 4y = -17\)[/tex]. To find the slope, we need to rewrite the equation in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
[tex]\[ 3x - 4y = -17 \][/tex]
[tex]\[ -4y = -3x - 17 \][/tex]
[tex]\[ y = \frac{3}{4}x + \frac{17}{4} \][/tex]
From this, we can see that the slope [tex]\(m\)[/tex] is [tex]\(\frac{3}{4}\)[/tex].
2. Recognize that parallel lines have the same slope:
Any line parallel to [tex]\(3x - 4y = -17\)[/tex] must have the same slope, which is [tex]\(\frac{3}{4}\)[/tex].
3. Write the equation of the new line:
The general form of the line parallel to the given line will be:
[tex]\[ 3x - 4y = C \][/tex]
where [tex]\(C\)[/tex] is a constant that we need to find.
4. Substitute the point [tex]\((-3, 2)\)[/tex] into the new equation to solve for [tex]\(C\)[/tex]:
[tex]\[ 3(-3) - 4(2) = C \][/tex]
[tex]\[ -9 - 8 = C \][/tex]
[tex]\[ C = -17 \][/tex]
5. Form the equation of the parallel line:
Substitute [tex]\(C\)[/tex] back into the general form:
[tex]\[ 3x - 4y = -17 \][/tex]
Therefore, the equation of the line that is parallel to the line [tex]\(3x - 4y = -17\)[/tex] and passes through the point [tex]\((-3, 2)\)[/tex] is:
[tex]\[ 3x - 4y = -17 \][/tex]
So, the correct answer from the given options is:
[tex]\[ \boxed{3x - 4y = -17} \][/tex]
