To determine the equation of the line that is parallel to the given line and passes through the point [tex]\((-3, 2)\)[/tex], we need to follow these steps:
1. Identify the slope of the given line:
The given line is [tex]\(3x - 4y = -17\)[/tex].
The general form of a line equation is [tex]\(Ax + By = C\)[/tex]. The slope of this line can be determined from its coefficients [tex]\(A\)[/tex] and [tex]\(B\)[/tex] as:
[tex]\[
\text{slope} = -\frac{A}{B}
\][/tex]
For the given line [tex]\(3x - 4y = -17\)[/tex], the coefficients are [tex]\(A = 3\)[/tex] and [tex]\(B = -4\)[/tex]. Thus, the slope is:
[tex]\[
\text{slope} = -\frac{3}{-4} = \frac{3}{4}
\][/tex]
2. Find the line with the same slope from the options:
We need to find another line among the provided options that also has a slope of [tex]\(\frac{3}{4}\)[/tex]. Checking each option:
- For [tex]\(3x - 4y = -17\)[/tex], as we calculated, the slope is [tex]\(\frac{3}{4}\)[/tex].
- For [tex]\(3x - 4y = -20\)[/tex], the coefficients are the same as the first line, so the slope is also [tex]\(\frac{3}{4}\)[/tex].
- For [tex]\(4x + 3y = -2\)[/tex], the slope would be [tex]\( -\frac{4}{3} \)[/tex].
- For [tex]\(4x + 3y = -6\)[/tex], the slope would be [tex]\( -\frac{4}{3} \)[/tex].
We see that both [tex]\(3x - 4y = -17\)[/tex] and [tex]\(3x - 4y = -20\)[/tex] have the desired slope of [tex]\(\frac{3}{4}\)[/tex].
3. Select the equation and verify:
Since we need the equation of the line parallel to [tex]\(3x - 4y = -17\)[/tex] (which has the same slope [tex]\(\frac{3}{4}\)[/tex]), we can use it directly because parallel lines have identical slopes.
4. Form the equation of the line passing through [tex]\((-3, 2)\)[/tex]:
The equation of a line in slope-intercept form ([tex]\(y = mx + b\)[/tex]) needs to pass through the point [tex]\((-3, 2)\)[/tex], where [tex]\(m\)[/tex] is the slope.
Using the slope [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substituting [tex]\(m = \frac{3}{4}\)[/tex] and the point [tex]\((-3, 2)\)[/tex]:
[tex]\[
y - 2 = \frac{3}{4}(x + 3)
\][/tex]
Distributing [tex]\(\frac{3}{4}\)[/tex] on the right side:
[tex]\[
y - 2 = \frac{3}{4}x + \frac{9}{4}
\][/tex]
Adding 2 (which is [tex]\(\frac{8}{4}\)[/tex]) to both sides:
[tex]\[
y = \frac{3}{4}x + \frac{9}{4} + \frac{8}{4} = \frac{3}{4}x + \frac{17}{4}
\][/tex]
However, we started with the given line [tex]\(3x - 4y = -17\)[/tex]. The key here is derivatives:
[tex]\[
y = \frac{3}{4}(x + 3) + 2 \rightarrow y = \frac{n}{m} is the same for both lines
(3, -4, -17)
\][/tex]
After comparing all of this, it confirms the equation:
The line among the given options, which is parallel to [tex]\(3x - 4y = -17\)[/tex], is:
[tex]\[
3x - 4y = -17
\][/tex]
Thus, the equation of the line that is parallel to [tex]\(3x - 4y = -17\)[/tex] is indeed:
[tex]\[
3x - 4y = -17
\][/tex]
