To find the equation of the line that is parallel to a given line and passes through a specific point, we need to follow these steps:
1. Identify the slope of the given line. Lines that are parallel have the same slope.
2. Use the point-slope form of the equation of a line. This form is useful when we know a point through which the line passes and its slope.
First, let’s find the slope of the given line [tex]\(3x - 4y = -17\)[/tex].
1. Rearrange 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 see that the slope [tex]\(m\)[/tex] is [tex]\(\frac{3}{4}\)[/tex].
Now, we use the point-slope form of the equation of a line [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\((x_1, y_1)\)[/tex] is a point on the line. In this case, the point is [tex]\((-3, 2)\)[/tex].
[tex]\[ y - 2 = \frac{3}{4}(x + 3) \][/tex]
We can simplify this equation:
[tex]\[ y - 2 = \frac{3}{4}x + \frac{3}{4}(3) \][/tex]
[tex]\[ y - 2 = \frac{3}{4}x + \frac{9}{4} \][/tex]
[tex]\[ y = \frac{3}{4}x + \frac{9}{4} + 2 \][/tex]
[tex]\[ y = \frac{3}{4}x + \frac{9}{4} + \frac{8}{4} \][/tex]
[tex]\[ y = \frac{3}{4}x + \frac{17}{4} \][/tex]
Now we convert this back to the standard form [tex]\(Ax + By = C\)[/tex]:
1. Multiply every term by 4 to clear the fractions:
[tex]\[ 4y = 3x + 17 \][/tex]
[tex]\[ 3x - 4y = -17 \][/tex]
So, the simplified equation of the line parallel to [tex]\(3x - 4y = -17\)[/tex] that passes through the point [tex]\((-3, 2)\)[/tex] turns out to be simpler than expected.
Thus, the required line equation is [tex]\(3x - 4y = -17\)[/tex].
However exact result may take is self-reflection based on presumed passed points and following accordance of option may infer refined calculations.
