To determine the equation of a line passing through a specific point with a given slope, we use the point-slope form of the equation of a line. The point-slope form is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line, and [tex]\(m\)[/tex] is the slope.
In this problem:
- The given point [tex]\((x_1, y_1)\)[/tex] is [tex]\((4, -6)\)[/tex].
- The slope [tex]\(m\)[/tex] is [tex]\(-\frac{3}{4}\)[/tex].
Plugging the given point and slope into the point-slope form, we get:
[tex]\[ y - (-6) = -\frac{3}{4}(x - 4) \][/tex]
[tex]\[ y + 6 = -\frac{3}{4}(x - 4) \][/tex]
Next, we can distribute the slope [tex]\(-\frac{3}{4}\)[/tex] on the right side:
[tex]\[ y + 6 = -\frac{3}{4}x + \left(-\frac{3}{4} \cdot -4\right) \][/tex]
[tex]\[ y + 6 = -\frac{3}{4}x + 3 \][/tex]
Now, isolate [tex]\(y\)[/tex] by subtracting 6 from both sides:
[tex]\[ y = -\frac{3}{4}x + 3 - 6 \][/tex]
[tex]\[ y = -\frac{3}{4}x - 3 \][/tex]
So, the equation of the line is:
[tex]\[ y = -\frac{3}{4}x - 3 \][/tex]
Comparing against the options:
1. [tex]\( y = -\frac{3}{4} x - 3 \)[/tex]
2. [tex]\( y = -\frac{3}{4} x - 6 \)[/tex]
3. [tex]\( y = -3 x - \frac{3}{4} \)[/tex]
4. [tex]\( y = -8 x - \frac{3}{4} \)[/tex]
The correct equation is [tex]\( y = -\frac{3}{4} x - 3 \)[/tex], which corresponds to option 1. Thus, the correct choice is:
[tex]\[ \boxed{1} \][/tex]
