To solve this problem, we need to determine the equation of a new route that is parallel to the given old route and passes through a specific point [tex]\((Q, P)\)[/tex].
The equation of the old route is given by:
[tex]\[ y = \frac{2}{5}x - 4 \][/tex]
### Step 1: Identify the Slope of the Old Route
The slope-intercept form of a line is given by:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
From the given equation:
[tex]\[ y = \frac{2}{5}x - 4 \][/tex]
we can identify the slope [tex]\( m \)[/tex] as:
[tex]\[ m = \frac{2}{5} \][/tex]
### Step 2: Determine the Slope of the New Route
Since the new route is parallel to the old route, it will have the same slope. Therefore, the slope of the new route is also:
[tex]\[ m = \frac{2}{5} \][/tex]
### Step 3: Write the Equation of the New Route in Point-Slope Form
The point-slope form of a line's equation that passes through a point [tex]\((x_1, y_1)\)[/tex] with slope [tex]\( m \)[/tex] is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Substituting the slope [tex]\( m = \frac{2}{5} \)[/tex] and the point [tex]\((Q, P)\)[/tex] into the point-slope form:
[tex]\[ y - P = \frac{2}{5}(x - Q) \][/tex]
### Step 4: Match with Given Options
Now, compare this with the given options:
1. [tex]\( y - Q = -\frac{5}{2}(x - P) \)[/tex]
2. [tex]\( y - Q = \frac{2}{5}(x - P) \)[/tex]
3. [tex]\( y - P = -\frac{5}{2}(x - Q) \)[/tex]
4. [tex]\( y - P = \frac{2}{5}(x - Q) \)[/tex]
We see that our derived equation [tex]\( y - P = \frac{2}{5}(x - Q) \)[/tex] matches option 4.
Therefore, the correct equation of the new route is:
[tex]\[ \boxed{4} \][/tex]
