To find the equation of the new route that is parallel to the old route and passes through the point [tex]\((Q, P)\)[/tex], let’s follow these steps:
1. Identify the slope of the old route:
The old route is given by the equation [tex]\(y = \frac{2}{5}x - 4\)[/tex]. This equation is in the form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope. Therefore, the slope of the old route is [tex]\(\frac{2}{5}\)[/tex].
2. Determine the slope of the new route:
Since we need the new route to be parallel to the old route, it must have the same slope. Therefore, the slope of the new route is also [tex]\(\frac{2}{5}\)[/tex].
3. Formulate the equation of the new route:
The point-slope form of a line's equation is given by:
[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.
For the new route, the slope [tex]\(m\)[/tex] is [tex]\(\frac{2}{5}\)[/tex], and it must pass through the point [tex]\((Q, P)\)[/tex]. Substituting [tex]\(m = \frac{2}{5}\)[/tex], [tex]\(x_1 = Q\)[/tex], and [tex]\(y_1 = P\)[/tex] into the point-slope form, we get:
[tex]\[
y - P = \frac{2}{5}(x - Q)
\][/tex]
4. Select the correct option:
Reviewing the provided options:
- [tex]\(y - Q = -\frac{5}{2}(x - P)\)[/tex]
- [tex]\(y - Q = \frac{2}{5}(x - P)\)[/tex]
- [tex]\(y - P = -\frac{5}{2}(x - Q)\)[/tex]
- [tex]\(y - P = \frac{2}{5}(x - Q)\)[/tex]
The correct form should match [tex]\(y - P = \frac{2}{5}(x - Q)\)[/tex]. Therefore, the correct equation for the new route is:
[tex]\[
y - P = \frac{2}{5}(x - Q)
\][/tex]
Thus, the correct option is:
[tex]\( \boxed{4} \)[/tex]
