To determine 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 need to follow these steps:
1. Identify the slope of the given line:
The equation of the line is [tex]\(3x - 4y = -17\)[/tex]. We rewrite this equation in slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
[tex]\[
3x - 4y = -17 \implies -4y = -3x - 17 \implies y = \frac{3}{4}x + \frac{17}{4}
\][/tex]
Therefore, the slope ([tex]\(m\)[/tex]) of the given line is [tex]\(\frac{3}{4}\)[/tex].
2. Use the point-slope form to write the equation of the new line:
Since the new line must be parallel to the given line, it will have the same slope, [tex]\(\frac{3}{4}\)[/tex]. We use the point-slope form of a line's equation, [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\((x_1, y_1)\)[/tex] is the point the line passes through.
Given point: [tex]\((-3, 2)\)[/tex] and slope [tex]\(\frac{3}{4}\)[/tex]
[tex]\[
y - 2 = \frac{3}{4}(x + 3)
\][/tex]
3. Simplify the equation:
Simplify the right-hand side equation:
[tex]\[
y - 2 = \frac{3}{4}x + \frac{9}{4}
\][/tex]
Now, add 2 to both sides, but remember to convert 2 into a fraction with a denominator of 4:
[tex]\[
y - 2 + 2 = \frac{3}{4}x + \frac{9}{4} + \frac{8}{4} \implies y = \frac{3}{4}x + \frac{17}{4}
\][/tex]
4. Convert the equation to standard form:
We convert [tex]\(y = \frac{3}{4}x + \frac{17}{4}\)[/tex] to the standard form [tex]\(Ax + By = C\)[/tex], where A, B, C are integers.
Multiply through by 4 to clear the denominators:
[tex]\[
4y = 3x + 17
\][/tex]
Move all the terms to one side to form an equation similar to the given ones:
[tex]\[
3x - 4y = -17
\][/tex]
5. Identify required constants:
Find the parallel line equation from the given options, note that coefficients for x and y should be the same, and the constant should be different. Among the given options, the equation [tex]\(3x - 4y = -20\)[/tex] satisfies these conditions with different constants.
Therefore, the equation of the line that is parallel to [tex]\(3x - 4y = -17\)[/tex] and passes through the point [tex]\((-3, 2)\)[/tex] is:
[tex]\[
3x - 4y = -20
\][/tex]
