To find the equation of the line that passes through the point [tex]\((6,14)\)[/tex] and is parallel to the line given by the equation [tex]\(y = -\frac{4}{3}x - 1\)[/tex], follow these steps:
1. Identify the slope of the given line:
The equation of the line is [tex]\(y = -\frac{4}{3}x - 1\)[/tex]. This equation is in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope. Thus, the slope ([tex]\(m\)[/tex]) of the given line is [tex]\(-\frac{4}{3}\)[/tex].
2. Determine the slope of the new, parallel line:
Lines that are parallel have the same slope. Therefore, the slope of the new line is also [tex]\(-\frac{4}{3}\)[/tex].
3. Use the point-slope form to find the equation of the new line:
The point-slope form of a line's equation 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 case, the line must pass through [tex]\((6, 14)\)[/tex] and has a slope of [tex]\(-\frac{4}{3}\)[/tex].
4. Plug in the known values:
[tex]\[
y - 14 = -\frac{4}{3}(x - 6)
\][/tex]
5. Distribute the slope and simplify:
[tex]\[
y - 14 = -\frac{4}{3}x + \frac{4}{3} \cdot 6
\][/tex]
[tex]\[
y - 14 = -\frac{4}{3}x + 8
\][/tex]
6. Solve for [tex]\(y\)[/tex] (convert to slope-intercept form):
[tex]\[
y = -\frac{4}{3}x + 8 + 14
\][/tex]
[tex]\[
y = -\frac{4}{3}x + 22
\][/tex]
So, the equation of the line that passes through the point [tex]\((6,14)\)[/tex] and is parallel to the line [tex]\(y = -\frac{4}{3}x - 1\)[/tex] is:
[tex]\[
y = -\frac{4}{3}x + 22
\][/tex]
Hence, the correct answer is D. [tex]\(y = -\frac{4}{3}x + 22\)[/tex].
