To determine the equations representing the line that is parallel to [tex]\(3x - 4y = 7\)[/tex] and passes through the point [tex]\((-4, -2)\)[/tex], we need to follow these steps:
1. Find the slope of the given line:
- The standard form of the line is [tex]\(3x - 4y = 7\)[/tex].
- Convert it to the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
[tex]\[
3x - 4y = 7 \implies -4y = -3x + 7 \implies y = \frac{3}{4}x - \frac{7}{4}
\][/tex]
- The slope [tex]\(m\)[/tex] of the line is [tex]\(\frac{3}{4}\)[/tex].
2. Determine the slope of the parallel line:
- Lines that are parallel have the same slope.
- Therefore, the slope of the line parallel to [tex]\(3x - 4y = 7\)[/tex] is also [tex]\(\frac{3}{4}\)[/tex].
3. Use the point-slope form to find the equation of the parallel line:
- We know the slope [tex]\(m = \frac{3}{4}\)[/tex] and the line passes through the point [tex]\((-4, -2)\)[/tex].
- The point-slope form of a line equation is [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\((x_1, y_1)\)[/tex] is a point on the line.
[tex]\[
y - (-2) = \frac{3}{4}(x - (-4)) \implies y + 2 = \frac{3}{4}(x + 4)
\][/tex]
4. Simplify and identify valid forms:
- The point-slope form directly gives us one of the options:
[tex]\[
y + 2 = \frac{3}{4}(x + 4)
\][/tex]
- Convert it to slope-intercept form:
[tex]\[
y + 2 = \frac{3}{4}x + 3 \implies y = \frac{3}{4}x + 1
\][/tex]
Now we compare these forms with the given options:
1. [tex]\(y = -\frac{3}{4}x + 1\)[/tex] (incorrect, wrong slope)
2. [tex]\(3x - 4y = -4\)[/tex] (correct, converting [tex]\(y = \frac{3}{4}x + 1\)[/tex] back to standard form):
[tex]\[
y = \frac{3}{4}x + 1 \implies 4y = 3x + 4 \implies 3x - 4y = -4
\][/tex]
3. [tex]\(4x - 3y = -3\)[/tex] (incorrect, wrong slope)
4. [tex]\(y - 2 = -\frac{3}{4}(x - 4)\)[/tex] (incorrect, wrong slope)
5. [tex]\(y + 2 = \frac{3}{4}(x + 4)\)[/tex] (correct, direct match from point-slope form)
The correct options are:
- [tex]\(3x - 4y = -4\)[/tex]
- [tex]\(y + 2 = \frac{3}{4}(x + 4)\)[/tex]
